The roots of the equation x² - 7x + 12 = 0 are: | Mathematics MCQ | Examoogle

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The roots of the equation x² - 7x + 12 = 0 are:

$3, 4$

$3, -4$

$2, 6$

$1, 12$

Correct Answer

📐 MA • Math Direct FactorizationMathematicsAlgebra

Option A

$3, 4$

Quick Summary:

Find two numbers whose product is 12 and sum is 7. Clearly, 3 and 4 satisfy both conditions ( $3 \times 4 = 12$ and $3 + 4 = 7$ ), so the roots are the opposite signs: 3 and 4.

📐MAMath SolutionDirect Factorization

📋 Given

Quadratic equation x² - 7x + 12 = 0

🔢 Formula Used

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac​​

⚡ Exam Hall Shortcut / Speed Trick

Find two numbers whose product is 12 and sum is 7. Clearly, 3 and 4 satisfy both conditions ( $3 \times 4 = 12$ and $3 + 4 = 7$ ), so the roots are the opposite signs: 3 and 4.

⚠️ Common Student Trap / Pitfall

Students often confuse the signs of the roots when factoring or forget that roots of $(x - p)(x - q) = 0$ are $p$ and $q$ , leading to selecting negative roots.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Identify coefficients

Compare the given equation $x^2 - 7x + 12 = 0$ with the standard quadratic form $ax^2 + bx + c = 0$ to get $a = 1$ , $b = -7$ , and $c = 12$ .

a=1,b=−7,c=12a = 1, b = -7, c = 12a=1,b=−7,c=12

Apply factorization

Split the middle term $-7x$ into $-3x$ and $-4x$ such that their product is $12x^2$ and sum is $-7x$ . Rewrite the equation as $x^2 - 3x - 4x + 12 = 0$ .

x(x−3)−4(x−3)=0x(x - 3) - 4(x - 3) = 0x(x−3)−4(x−3)=0

Solve for x

Factor out $(x - 3)$ to get $(x - 3)(x - 4) = 0$ , which yields the roots $x = 3$ and $x = 4$ .

x=3 or x=4x = 3 \text{ or } x = 4x=3 or x=4

✅

A is correct because the roots of the equation $x^2 - 7x + 12 = 0$ are 3 and 4.

Core Concepts Used

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Quadratic Equations Factorization Method Roots of Polynomials

💡 EXAM TIP

Understanding the relationship between roots and coefficients (Vieta's formulas) helps quickly verify solutions in algebra and higher-order polynomial equations.

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