NATURE OF QUADRATIC ROOTS

 Quadratic Roots

1. Introduction

A quadratic equation is any equation of the form

$$ax^2 + bx + c = 0,\quad \text{with } a \neq 0.$$

Its solutions (or roots) are the values of $x$ that satisfy the equation. These roots can be real or complex depending on the values of the coefficients $a$, $b$, and $c$. In solving quadratics, we often use the quadratic formula, analyze the discriminant, and sometimes recognize perfect square trinomials.

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2. Quadratic Formula and the Discriminant

The Quadratic Formula

For a quadratic equation $ax^2 + bx + c = 0$, the quadratic formula gives the roots as:

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

The Discriminant

The expression under the square root, $D = b^2 - 4ac$, is known as the discriminant. It determines the nature of the roots:

• $D > 0$: Two distinct real roots.
• $D = 0$: One real repeated (double) root.
• $D < 0$: Two complex conjugate roots.

Example:
Consider the quadratic $x^2 - 4x + 3 = 0$. Here, $a=1$, $b=-4$, and $c=3$. The discriminant is

$$D = (-4)^2 - 4(1)(3) = 16 - 12 = 4 \quad (D > 0),$$

so there are two distinct real roots.

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3. Sum and Product of Roots

Let the two roots be $\alpha$ and $\beta$. If the quadratic factors as

$$a(x - \alpha)(x - \beta) = ax^2 - a(\alpha + \beta)x + a\alpha\beta,$$

by comparing coefficients with $ax^2 + bx + c$, we obtain:

• Sum of the roots:

  $$\alpha + \beta = -\frac{b}{a}$$

• Product of the roots:

  $$\alpha\beta = \frac{c}{a}.$$

Derivation Using the Quadratic Formula

Starting from the quadratic formula:

$$\alpha,\, \beta = \frac{-b \pm \sqrt{b^2-4ac}}{2a},$$

their sum is

$$\alpha + \beta = \frac{-b + \sqrt{b^2-4ac}}{2a} + \frac{-b - \sqrt{b^2-4ac}}{2a} = \frac{-2b}{2a} = -\frac{b}{a}.$$

For the product, note that

$$\alpha\beta = \left(\frac{-b + \sqrt{b^2-4ac}}{2a}\right)\left(\frac{-b - \sqrt{b^2-4ac}}{2a}\right) = \frac{b^2 - (b^2-4ac)}{4a^2} = \frac{4ac}{4a^2} = \frac{c}{a}.$$

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4. Perfect Square Trinomials

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It takes the form:

$$(px + q)^2 = p^2x^2 + 2pqx + q^2.$$

For a quadratic $ax^2 + bx + c$ to be a perfect square, the following condition must hold:

$$b^2 = 4ac.$$

Example:
Consider $x^2 + 6x + 9$. Here, $b^2 = 36$ and $4ac = 4(1)(9)=36$. Since $b^2 = 4ac$, the trinomial is perfect. In fact,

$$x^2 + 6x + 9 = (x + 3)^2.$$

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5. Worked JAMB Exam Examples

Below are five worked examples modeled on JAMB exam questions:

Example 1:

Question: Solve $x^2 - 5x + 6 = 0$ and find the sum and product of the roots.

Solution:

1. Factorization: $$x^2 - 5x + 6 = (x - 2)(x - 3) = 0.$$
2. Roots: $x = 2$ and $x = 3$.
3. Sum: $2 + 3 = 5$ (which is $-\frac{-5}{1}$).
4. Product: $2 \times 3 = 6$ (which is $\frac{6}{1}$).

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Example 2:

Question: Solve $2x^2 - 8x + 8 = 0$.

Solution:

1. Quadratic Formula: $$x = \frac{8 \pm \sqrt{(-8)^2 - 4(2)(8)}}{2 \times 2} = \frac{8 \pm \sqrt{64 - 64}}{4} = \frac{8 \pm 0}{4}.$$
2. Root: $x = 2$ (a repeated root).
3. Sum and Product:
   • Sum: $2 + 2 = 4 = -\frac{-8}{2}$.
   • Product: $2 \times 2 = 4 = \frac{8}{2}$.

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Example 3:

Question: Solve $x^2 + 2x + 5 = 0$.

Solution:

1. Compute the Discriminant: $$D = 2^2 - 4(1)(5) = 4 - 20 = -16.$$
2. Since $D < 0$: The equation has two complex roots.
3. Quadratic Formula: $$x = \frac{-2 \pm \sqrt{-16}}{2} = \frac{-2 \pm 4i}{2} = -1 \pm 2i.$$

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Example 4:

Question: Determine whether $9x^2 + 12x + 4$ is a perfect square trinomial.

Solution:

1. Check the condition: $$b^2 = 12^2 = 144 \quad \text{and} \quad 4ac = 4(9)(4) = 144.$$
2. Since $b^2 = 4ac$: It is a perfect square.
3. Express as a square: $$9x^2 + 12x + 4 = (3x+2)^2.$$

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Example 5:

Question: For the quadratic $3x^2 - 12x + 9 = 0$, find the roots and verify the sum and product formulas.

Solution:

1. Divide the equation by 3: $$x^2 - 4x + 3 = 0.$$
2. Factorization: $$(x-1)(x-3)=0 \quad \Rightarrow \quad x=1 \text{ or } x=3.$$
3. Sum of Roots: $1 + 3 = 4 = -\frac{-12}{3}$.
4. Product of Roots: $1 \times 3 = 3 = \frac{9}{3}$.

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6. JAMB Practice Questions (with Options)

Here are 10 practice questions similar to those encountered in JAMB examinations:

Q1. Solve for $x$:

$$x^2 - 3x + 2 = 0.$$

A) $x = 1, 2$
B) $x = -1, -2$
C) $x = 2 \pm i$
D) $x = 1 \pm i$

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Q2. The discriminant of the quadratic equation $2x^2 + 4x + k = 0$ is zero. Find $k$.
A) 2
B) 4
C) 3
D) 8

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Q3. If the roots of $ax^2 + bx + c = 0$ are $\alpha$ and $\beta$, then the sum of the roots is:
A) $\frac{b}{a}$
B) $-\frac{b}{a}$
C) $\frac{c}{a}$
D) $-\frac{c}{a}$

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Q4. For the quadratic $x^2 + 6x + 9 = 0$, the nature of the roots is:
A) Real and distinct
B) Real and equal
C) Complex conjugates
D) Irrational

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Q5. Identify the value of $c$ for which $x^2 - 10x + c$ becomes a perfect square trinomial.
A) 16
B) 25
C) 36
D) 49

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Q6. Solve the quadratic equation using the quadratic formula:

$$3x^2 - 5x + 2 = 0.$$

A) $x = 1, \frac{2}{3}$
B) $x = \frac{2}{3}, 1$
C) $x = -1, -\frac{2}{3}$
D) $x = 1, -\frac{2}{3}$

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Q7. If the product of the roots of a quadratic equation $ax^2+bx+c=0$ is 3, then

$$\alpha\beta = \frac{c}{a} = 3.$$

Which of the following could be a correct form of the equation?
A) $x^2 - 5x + 3 = 0$
B) $2x^2 - 4x + 3 = 0$
C) $x^2 + 3x + 3 = 0$
D) $2x^2 + 3x + 3 = 0$

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Q8. Find the roots of the equation $x^2 + 2x + 5 = 0$.
A) $-1 \pm 2i$
B) $1 \pm 2i$
C) $-1 \pm i$
D) $1 \pm i$

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Q9. The quadratic equation $4x^2 + kx + 9 = 0$ has no real roots. Which condition on $k$ is correct?
A) $k^2 < 144$
B) $k^2 > 144$
C) $k^2 < 0$
D) $k^2 = 144$

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Q10. For the equation $x^2 - 4x + 4 = 0$, what is the value of the discriminant and what does it indicate?
A) $D=0$, indicating two distinct real roots
B) $D=0$, indicating one repeated real root
C) $D>0$, indicating two distinct real roots
D) $D<0$, indicating complex roots

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Final Remarks

This note has detailed the derivation and interpretation of the quadratic formula, the significance of the discriminant, and the use of Vieta’s formulas to find the sum and product of the roots. In addition, the explanation of perfect square trinomials aids in recognizing when a quadratic can be easily factored. The worked examples are representative of JAMB exam problems, and the practice questions are designed to reinforce these key concepts.

This comprehensive guide should serve as a useful reference for understanding quadratic equations in both theory and exam practice 

JAMB Practice Questions on Quadratic Equations

1. Solve for $x$ in the quadratic equation $x^2 - 7x + 12 = 0$.
A) $x = 3, 4$
B) $x = 2, 6$
C) $x = 1, 12$
D) $x = 5, 7$

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2. Find the sum and product of the roots of the equation $3x^2 - 5x + 2 = 0$.
A) Sum = $\frac{5}{3}$, Product = $\frac{2}{3}$
B) Sum = $\frac{2}{3}$, Product = $\frac{5}{3}$
C) Sum = $5$, Product = $2$
D) Sum = $3$, Product = $5$

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3. If the discriminant of a quadratic equation is negative, then the roots are:
A) Real and distinct
B) Real and equal
C) Complex (imaginary)
D) Rational and different

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4. What is the discriminant of the quadratic equation $x^2 - 6x + 9 = 0$?
A) 9
B) 0
C) -9
D) 6

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5. Which of the following quadratic equations is a perfect square trinomial?
A) $x^2 + 8x + 16 = 0$
B) $x^2 + 5x + 10 = 0$
C) $2x^2 + 3x + 4 = 0$
D) $x^2 - 4x + 5 = 0$

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6. The equation $2x^2 - 3x + 1 = 0$ has roots given by the quadratic formula. What is the correct solution?
A) $x = 1, \frac{1}{2}$
B) $x = 3, 2$
C) $x = -1, -\frac{1}{2}$
D) $x = 2, \frac{3}{2}$

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7. If one root of the quadratic equation $x^2 + kx + 12 = 0$ is 3, find the value of $k$.
A) -7
B) 7
C) -5
D) 5

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8. The product of the roots of the quadratic equation $4x^2 - 10x + 6 = 0$ is:
A) $6$
B) $\frac{6}{4}$
C) $\frac{3}{2}$
D) $4$

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9. For what value of $k$ does the quadratic equation $x^2 - 4x + k = 0$ have equal roots?
A) 4
B) 2
C) 1
D) 8

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10. Which of the following quadratic equations has no real roots?
A) $x^2 - 5x + 6 = 0$
B) $x^2 + 4x + 5 = 0$
C) $x^2 - 2x - 3 = 0$
D) $x^2 - 3x + 2 = 0$

Let me know if you need solutions or modifications!