The minimum value of $ 8x + 14 + 3x^2 $ is: | Mathematics MCQ | Examoogle

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The minimum value of $8x + 14 + 3x^2$ is:

$8\frac{2}{3}$

$-\frac{28}{3}$

$-8\frac{2}{3}$

$\frac{28}{3}$

Correct Answer

📐 MA • Math Quadratic CalculusMathematicsPrevious Year Paper 2024

Option A

$8\frac{2}{3}$

Quick Summary:

For any quadratic $ax^2 + bx + c$ with $a > 0$ , the minimum value is reached at $x = -b / (2a)$ . Substituting $x = -b / (2a)$ into the expression gives the minimum value directly.

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📐MAMath SolutionQuadratic Calculus

📋 Given

A quadratic expression: 8x + 14 + 3x²

🔢 Formula Used

f(x)=ax2+bx+cf(x) = ax^2 + bx + cf(x)=ax2+bx+c, Minimum value=c−b24a\text{Minimum value} = c - \frac{b^2}{4a}Minimum value=c−4ab2​

⚡ Exam Hall Shortcut / Speed Trick

For any quadratic $ax^2 + bx + c$ with $a > 0$ , the minimum value is reached at $x = -b / (2a)$ . Substituting $x = -b / (2a)$ into the expression gives the minimum value directly.

⚠️ Common Student Trap / Pitfall

Confusing the location of the minimum ( $x = -b/2a$ ) with the value of the minimum ( $f(-b/2a)$ ).

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Rearrange the expression

Standardize the quadratic expression $8x + 14 + 3x^2$ into the form $ax^2 + bx + c$ .

3x2+8x+143x^2 + 8x + 143x2+8x+14

Identify coefficients

For the quadratic $3x^2 + 8x + 14$ , identify the values of $a$ , $b$ , and $c$ .

a=3,b=8,c=14a = 3, b = 8, c = 14a=3,b=8,c=14

Calculate the minimum value

Use the formula for the minimum value of a quadratic: $c - \frac{b^2}{4a}$ .

14−824×3=14−6412=14−16314 - \frac{8^2}{4 \times 3} = 14 - \frac{64}{12} = 14 - \frac{16}{3}14−4×382​=14−1264​=14−316​

Simplify the result

Perform the final subtraction to find the value.

42−163=263\frac{42 - 16}{3} = \frac{26}{3}342−16​=326​

✅

B is correct because the minimum value of the quadratic $3x^2 + 8x + 14$ is calculated as $14 - \frac{64}{12} = \frac{26}{3}$ . Note: Although the provided official key says A, the mathematical evaluation yields 26/3, which corresponds to option B.

Core Concepts Used

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Quadratic Minima Vertex of Parabola Algebraic Simplification

💡 EXAM TIP

This concept of finding the vertex of a parabola is fundamental in coordinate geometry and optimization problems across physics (e.g., projectile motion) and calculus.

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