Two Equal Roots Condition

Introduction

Key Concepts

Understanding Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable \(x\), with the general form:

$$ax^2 + bx + c = 0$$

where \(a\), \(b\), and \(c\) are coefficients, and \(a \neq 0\). The solutions to this equation are the roots, which can be real or complex numbers.

Discriminant and Its Role

The discriminant of a quadratic equation, denoted as \(D\), is given by:

$$D = b^2 - 4ac$$

The discriminant determines the nature of the roots:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), the equation has two equal real roots.
• If \(D < 0\), the equation has two complex conjugate roots.

Condition for Two Equal Roots

For a quadratic equation to have two equal roots, the discriminant must be zero:

$$b^2 - 4ac = 0$$

Solving for the condition:

$$b^2 = 4ac$$

This equation signifies that the quadratic equation touches the x-axis at exactly one point, indicating a repeated root.

Finding the Equal Root

When the discriminant is zero, the root can be found using the formula:

$$x = \frac{-b}{2a}$$

This is derived from the quadratic formula:

$$x = \frac{-b \pm \sqrt{D}}{2a}$$

Substituting \(D = 0\) simplifies the expression to a single root.

Graphical Interpretation

Graphically, a quadratic equation with two equal roots represents a parabola that just touches the x-axis at the vertex. The coordinates of the vertex are \(\left( \frac{-b}{2a}, 0 \right)\).

Example: Consider the equation \(x^2 - 4x + 4 = 0\).

Here, \(a = 1\), \(b = -4\), and \(c = 4\).

Calculating the discriminant:

$$D = (-4)^2 - 4(1)(4) = 16 - 16 = 0$$

Since \(D = 0\), the equation has two equal roots:

$$x = \frac{-(-4)}{2(1)} = \frac{4}{2} = 2$$

Thus, the root is \(x = 2\), and the graph touches the x-axis at \((2, 0)\).

Applications of Two Equal Roots Condition

Understanding the two equal roots condition is crucial in various applications:

• Physics: Determining moments of equilibrium where objects balance perfectly.
• Engineering: Analyzing stress points in structures where forces are balanced.
• Economics: Modeling scenarios where supply equals demand at a specific price point.

Deriving the Equal Roots Condition

Starting with the general quadratic equation:

$$ax^2 + bx + c = 0$$

The roots are given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

For the roots to be equal:

$$\sqrt{b^2 - 4ac} = 0$$

Squaring both sides:

$$b^2 - 4ac = 0$$

Thus, the condition for two equal roots is:

$$b^2 = 4ac$$

Examples Illustrating Two Equal Roots

Example 1: Solve the equation \(2x^2 + 4x + 2 = 0\) and determine if it has two equal roots.

Calculating the discriminant:

$$D = 4^2 - 4(2)(2) = 16 - 16 = 0$$

Since \(D = 0\), the equation has two equal roots:

$$x = \frac{-4}{2(2)} = \frac{-4}{4} = -1$$

The root is \(x = -1\).

Example 2: Determine if the equation \(3x^2 - 6x + 3 = 0\) has two equal roots.

Calculating the discriminant:

$$D = (-6)^2 - 4(3)(3) = 36 - 36 = 0$$

Since \(D = 0\), the equation has two equal roots:

$$x = \frac{6}{2(3)} = \frac{6}{6} = 1$$

The root is \(x = 1\).

Solving Quadratic Equations with Two Equal Roots

To solve a quadratic equation that meets the two equal roots condition:

1. Ensure that the discriminant \(D = b^2 - 4ac\) is zero.
2. Apply the formula \(x = \frac{-b}{2a}\) to find the equal roots.

Example: Solve \(5x^2 + 10x + 5 = 0\).

Calculate the discriminant:

$$D = 10^2 - 4(5)(5) = 100 - 100 = 0$$

Since \(D = 0\), the equation has two equal roots:

$$x = \frac{-10}{2(5)} = \frac{-10}{10} = -1$$

The root is \(x = -1\).

Graphical Representation of Two Equal Roots

The graph of a quadratic equation with two equal roots is a parabola that touches the x-axis at its vertex. The vertex form of the quadratic equation is:

$$y = a(x - h)^2 + k$$

For two equal roots, \(k = 0\) and the vertex is at \((h, 0)\).

Example: Graph the equation \(x^2 - 2x + 1 = 0\).

Rewriting in vertex form:

$$y = (x - 1)^2$$

The vertex is at \((1, 0)\), and the parabola touches the x-axis at this point, indicating two equal roots at \(x = 1\).

Real-World Problems Involving Two Equal Roots

Consider the scenario where a ball is thrown vertically upwards, and we want to find the moment when it returns to the same height. Using the two equal roots condition can help determine the exact time when the ball is at its peak height.

Problem: A ball is thrown upwards with an initial velocity of 20 m/s from the ground. Its height \(h\) in meters at time \(t\) seconds is given by:

$$h = -5t^2 + 20t$$

To find when the ball returns to the ground, set \(h = 0\):

$$-5t^2 + 20t = 0$$

Factorizing:

$$-5t(t - 4) = 0$$

Thus, \(t = 0\) or \(t = 4\). The ball is on the ground at \(t = 0\) and \(t = 4\) seconds. At \(t = 2\) seconds, the ball reaches its peak, where the roots are equal.

Advanced Concepts

Multiplicity of Roots

When a quadratic equation has two equal roots, it implies that the root has a multiplicity of two. In algebraic terms, the equation can be expressed as:

$$a(x - r)^2 = 0$$

where \(r\) is the repeated root. This form emphasizes the double occurrence of the root \(r\).

Expanding the equation:

$$a(x^2 - 2rx + r^2) = 0$$

Comparing with the standard form \(ax^2 + bx + c = 0\), we identify:

$$b = -2ar$$

$$c = ar^2$$

Hence, the condition \(b^2 = 4ac\) holds, reinforcing the two equal roots scenario.

Deriving the Two Equal Roots Condition Using Vieta's Theorem

Vieta's Theorem relates the coefficients of a polynomial to the sums and products of its roots. For the quadratic equation:

$$ax^2 + bx + c = 0$$

Let the roots be \(r_1\) and \(r_2\). According to Vieta's Theorem:

• Sum of roots: \(r_1 + r_2 = -\frac{b}{a}\)
• Product of roots: \(r_1r_2 = \frac{c}{a}\)

For two equal roots, \(r_1 = r_2 = r\). Substituting into Vieta's formulas:

$$2r = -\frac{b}{a} \Rightarrow r = -\frac{b}{2a}$$

$$r^2 = \frac{c}{a} \Rightarrow \left(-\frac{b}{2a}\right)^2 = \frac{c}{a}$$

Simplifying:

$$\frac{b^2}{4a^2} = \frac{c}{a} \Rightarrow b^2 = 4ac$$

This derivation confirms the condition for two equal roots using Vieta's Theorem.

Exploring the Role of the Leading Coefficient

The leading coefficient \(a\) in a quadratic equation affects the direction and width of the parabola but does not alter the condition for two equal roots. Regardless of the value of \(a\), as long as the condition \(b^2 = 4ac\) is satisfied, the equation will have two equal roots.

Example: Compare the equations \(x^2 - 6x + 9 = 0\) and \(2x^2 - 12x + 18 = 0\).

For the first equation:

$$D = (-6)^2 - 4(1)(9) = 36 - 36 = 0$$

For the second equation:

$$D = (-12)^2 - 4(2)(18) = 144 - 144 = 0$$

Both equations satisfy \(D = 0\) and thus have two equal roots, despite different leading coefficients.

Advanced Problem-Solving Techniques

Solving quadratic equations with two equal roots often involves recognizing perfect square trinomials. Additionally, utilizing the vertex form of a parabola can simplify the process of identifying equal roots.

Example: Solve \(4x^2 + 12x + 9 = 0\).

First, check the discriminant:

$$D = 12^2 - 4(4)(9) = 144 - 144 = 0$$

Since \(D = 0\), there are two equal roots:

$$x = \frac{-12}{2(4)} = \frac{-12}{8} = -1.5$$

The root is \(x = -1.5\).

Alternatively, recognize the equation as a perfect square:

$$4x^2 + 12x + 9 = (2x + 3)^2 = 0$$

Setting the squared term to zero gives:

$$2x + 3 = 0 \Rightarrow x = -\frac{3}{2}$$

Complex Numbers and Equal Roots

When the discriminant \(D = 0\), the roots are real and equal. However, if we extend our discussion to complex numbers, the concept of multiplicity still applies. The repeated roots are identical in both their real and imaginary parts.

Example: Solve \(x^2 + 4x + 4 = 0\).

Calculating the discriminant:

$$D = 4^2 - 4(1)(4) = 16 - 16 = 0$$

Thus, the equation has two equal roots:

$$x = \frac{-4}{2(1)} = -2$$

In the complex plane, both roots are \(-2 + 0i\).

Intersection of Parabolas with Equal Roots

Consider two parabolas intersecting at a single point. The condition for them to have equal roots is that they share a common tangent at the point of intersection.

Example: Find the point where the parabolas \(y = x^2\) and \(y = 2x^2 - 4x + 2\) intersect with equal roots.

Set the equations equal to each other:

$$x^2 = 2x^2 - 4x + 2$$

Rearranging:

$$x^2 - 4x + 2 = 0$$

Calculating the discriminant:

$$D = (-4)^2 - 4(1)(2) = 16 - 8 = 8 \neq 0$$

Since \(D \neq 0\), the parabolas do not intersect with equal roots. To have equal roots, adjust the equations accordingly.

If the second equation is \(y = x^2\), both parabolas coincide, resulting in infinitely many roots. For exactly two equal roots, ensure that the discriminant of the resulting equation is zero.

Connection with the Vertex Form

The vertex form of a quadratic equation provides insights into the nature of its roots. The vertex is the point where the parabola changes direction, and if it lies on the x-axis, the equation has two equal roots.

Vertex form:

$$y = a(x - h)^2 + k$$

For two equal roots, set \(k = 0\):

$$y = a(x - h)^2$$

The vertex is at \((h, 0)\), and the root is \(x = h\).

Example: Convert \(y = 3x^2 - 6x + 3\) to vertex form.

Factor out the coefficient of \(x^2\):

$$y = 3\left(x^2 - 2x + 1\right) = 3(x - 1)^2$$

The vertex is at \((1, 0)\), indicating two equal roots at \(x = 1\).

Handling Equations with Leading Coefficient Zero

In cases where the leading coefficient \(a = 0\), the equation becomes linear, not quadratic. Hence, the two equal roots condition strictly applies to genuine quadratic equations where \(a \neq 0\).

Example: Analyze the equation \(0x^2 + 5x + 5 = 0\).

Since \(a = 0\), the equation simplifies to:

$$5x + 5 = 0$$

Solving for \(x\):

$$x = -1$$

This linear equation has a single root, not two equal roots.

Exploring Systems of Equations with Equal Roots

When dealing with systems of quadratic equations, the two equal roots condition can help identify points of tangency or coinciding solutions.

Example: Find the value of \(k\) for which the system below has two equal roots:

$$y = x^2 + kx + 1$$

$$y = 2x + 3$$

Setting the equations equal:

$$x^2 + kx + 1 = 2x + 3$$

Rearranging:

$$x^2 + (k - 2)x - 2 = 0$$

For two equal roots:

$$D = (k - 2)^2 - 4(1)(-2) = (k - 2)^2 + 8 = 0$$

Since \((k - 2)^2 \geq 0\) and \(8 > 0\), there are no real values of \(k\) that satisfy \(D = 0\). Hence, the system does not have two equal roots for any real \(k\).

Analyzing Complex Quadratic Equations

While the two equal roots condition typically refers to real roots, complex quadratic equations can also exhibit repeated roots in the complex plane.

Example: Solve \(x^2 + 2x + 1 = 0\).

Factorizing:

$$(x + 1)^2 = 0$$

Thus, the repeated root is \(x = -1\).

Even though the roots are real, the concept extends to complex numbers by considering multiplicity.

Implications in Calculus

In calculus, the two equal roots condition relates to the extrema of quadratic functions. The vertex of a parabola, where the derivative equals zero, corresponds to the repeated root when the parabola just touches the x-axis.

Example: Find the critical point of \(f(x) = 2x^2 - 8x + 8\).

First, find the derivative:

$$f'(x) = 4x - 8$$

Setting \(f'(x) = 0\):

$$4x - 8 = 0 \Rightarrow x = 2$$

Substituting \(x = 2\) into \(f(x)\):

$$f(2) = 2(2)^2 - 8(2) + 8 = 8 - 16 + 8 = 0$$

The vertex is at \((2, 0)\), indicating two equal roots at \(x = 2\).

Using the Quadratic Formula for Equal Roots

The quadratic formula is a versatile tool for finding the roots of any quadratic equation. When applying it to equations with two equal roots, the formula simplifies significantly.

Formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

For two equal roots (\(D = 0\)):

$$x = \frac{-b \pm 0}{2a} = \frac{-b}{2a}$$

This simplification underscores the uniqueness of the repeated root.

Example: Solve \(6x^2 + 12x + 6 = 0\) using the quadratic formula.

Calculating the discriminant:

$$D = 12^2 - 4(6)(6) = 144 - 144 = 0$$

Thus, the roots are:

$$x = \frac{-12}{2(6)} = \frac{-12}{12} = -1$$

The root is \(x = -1\).

Symmetric Properties of Quadratic Equations with Equal Roots

Quadratic equations with two equal roots exhibit symmetry about the axis of symmetry, which passes through the vertex. This symmetry is reflected in the equal distances of points on the parabola from the axis.

The axis of symmetry for the equation \(ax^2 + bx + c = 0\) is:

$$x = \frac{-b}{2a}$$

For two equal roots, this axis intersects the x-axis at the repeated root, ensuring symmetrical properties.

Example: Consider \(x^2 - 4x + 4 = 0\).

The axis of symmetry is:

$$x = \frac{4}{2(1)} = 2$$

The repeated root is at \(x = 2\), confirming the symmetry.

Exploring Quadratic Functions with Two Equal Roots

Quadratic functions with two equal roots have their vertex lying on the x-axis. This unique property affects the function's graph, derivative, and integral.

Graph: The parabola touches the x-axis at the vertex \((h, 0)\).

Derivative: The first derivative at the vertex is zero, indicating a minimum or maximum point.

Integral: The area under the curve from the vertex shows symmetry.

Example: Analyze \(f(x) = -x^2 + 6x - 9\).

Rewriting in vertex form:

$$f(x) = -(x^2 - 6x + 9) = -(x - 3)^2 + 0$$

The vertex is at \((3, 0)\), and the parabola opens downward, indicating a maximum point with a repeated root at \(x = 3\).

Connection with Derivatives and Tangents

In calculus, the condition for two equal roots corresponds to the tangent line to the parabola being parallel to the x-axis. The derivative at the point of tangency equals zero, aligning with the two equal roots condition.

Example: Find the tangent to the curve \(y = x^2 - 4x + 4\) at the vertex.

First, find the derivative:

$$y' = 2x - 4$$

At the vertex \(x = 2\):

$$y'(2) = 2(2) - 4 = 0$$

The tangent line is horizontal at this point, confirming the repeated root at \(x = 2\).

Investigating Higher-Degree Polynomials

While the two equal roots condition primarily pertains to quadratic equations, similar concepts apply to higher-degree polynomials. Repeated roots in higher-degree polynomials indicate multiple factors with the same root value.

Example: Consider the polynomial \(f(x) = (x - 1)^2(x + 2)\).

This polynomial has a repeated root at \(x = 1\) and a distinct root at \(x = -2\). The repeated root \(x = 1\) satisfies the two equal roots condition within the quadratic factor \((x - 1)^2\).

Exploring Equal Roots in Complex Plane

In the complex plane, the concept of two equal roots extends to the multiplicity of roots regardless of their real or imaginary components. Repeated complex roots behave similarly to their real counterparts in terms of multiplicity and factorization.

Example: Solve \(x^2 + 2x + 1 = 0\) in the complex plane.

Factorizing:

$$x^2 + 2x + 1 = (x + 1)^2 = 0$$

The repeated root is \(x = -1 + 0i\), which lies on the real axis.

Application in Optimization Problems

Optimization problems often involve finding maximum or minimum values, where the two equal roots condition indicates critical points. Quadratic functions, with their parabolic shapes, are frequently utilized in such scenarios.

Example: Determine the dimensions of a rectangle with maximum area under the constraint that its length equals its width.

Let the side length be \(x\). The area \(A\) is:

$$A = x^2$$

To maximize \(A\) under certain constraints, calculus can be applied. However, since \(A = x^2\) is always increasing for \(x > 0\), the maximum occurs at the boundary of the feasible region. In cases where constraints lead to a quadratic equation derived from setting the derivative to zero, the two equal roots condition can identify optimal dimensions.

Understanding Repeated Roots through Polynomial Division

Polynomial division can demonstrate the multiplicity of roots. Dividing a polynomial by \((x - r)\) repeatedly shows whether \(r\) is a single or repeated root.

Example: Factorize \(f(x) = x^3 - 3x^2 + 3x - 1\).

Using the Rational Root Theorem, test \(x = 1\):

$$f(1) = 1 - 3 + 3 - 1 = 0$$

Performing polynomial division or synthetic division with \(x - 1\):

$$f(x) = (x - 1)(x^2 - 2x + 1) = (x - 1)^3$$

The root \(x = 1\) has multiplicity three, illustrating repeated roots beyond the quadratic case.

Exploring Symmetric Functions with Equal Roots

Symmetric functions exhibit properties that remain invariant under variable substitution. In quadratic equations, symmetry around the axis of symmetry ensures equal distribution of roots.

Example: Analyze the symmetry in \(f(x) = x^2 - 6x + 9\).

The axis of symmetry is \(x = 3\). The repeated root \(x = 3\) lies on this axis, reflecting the equation's symmetric property.

Exploring the Impact of Coefficient Changes

Altering coefficients in a quadratic equation affects the roots' nature. Specifically, increasing or decreasing coefficients \(a\), \(b\), or \(c\) can transition the equation from having two distinct real roots to two equal roots or to complex roots.

Example: Modify the equation \(x^2 + 4x + 4 = 0\) to change its roots from equal to distinct.

Original equation has \(D = 0\). To have distinct roots, adjust \(c\) such that \(D > 0\). For instance, let \(c = 3\):

$$x^2 + 4x + 3 = 0$$

Calculating the discriminant:

$$D = 16 - 12 = 4 > 0$$

The roots are \(x = -1\) and \(x = -3\), now distinct.

Implications in Real-Life Scenarios

In real-life scenarios, the two equal roots condition can represent situations where two different factors converge at a single outcome. For instance, in finance, it could represent break-even points where revenue equals cost at a specific production level.

Example: A company's revenue and cost functions intersect at a single production level, indicating break-even with two equal roots.

If Revenue \(R(x) = 50x\) and Cost \(C(x) = 25x^2 + 50x + 25\), setting \(R(x) = C(x)\):

$$50x = 25x^2 + 50x + 25$$

Rearranging:

$$25x^2 + 25 = 0$$

Calculating discriminant:

$$D = 0^2 - 4(25)(25) = -2500 < 0$$

No real solutions, indicating no break-even points. To achieve two equal roots, adjust the cost function.

Comparison Table

Summary and Key Takeaways