CAT Quadratic formula - Practice Questions & MCQ

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5 Questions around this concept.

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If one root of the equation $px^{2}+qx+r=0$ , $p\neq 0$ is reciprocal of the other, then,

q = r

p = r

p = 0

q = 0

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If $\alpha ,\beta \\$ are roots $ax^{2}+bx+c=0\\$ of then fin the value of $\frac{1}{\left ( a\alpha +b \right )}+\frac{1}{\left (a\beta +b \right )}$

$\frac{bc}{a}$

$\frac{ac}{b}$

$\frac{b}{ac}$

$\frac{a}{bc}$

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Concepts Covered - 1

Quadratic formula

Introduction to Quadratic Formula:

The Quadratic Formula is a powerful tool used to find the roots (solutions) of a quadratic equation. It is derived from completing the square method and can be used to solve any quadratic equation in the form $\mathrm{ax^{2} + bx + c = 0}$ , where a, b, and c are constants and $\mathrm{a \neq 0.}$

Quadratic Equation:

A quadratic equation is a polynomial equation of degree 2. It can be written in the standard form ax² + bx + c = 0, where a, b, and c are coefficients, and x represents the variable.

Roots of a Quadratic Equation:

The roots of a quadratic equation are the values of x that satisfy the equation and make it true. A quadratic equation can have two distinct real roots, two identical real roots, or two complex roots.

Derivation of Quadratic Formula:

The quadratic formula is derived by applying the completing the square method to the generic quadratic equation $\mathrm{ax^{2} + bx + c = 0. }$ The formula is: $\mathrm{x = (-b \pm \sqrt(b^{2}- 4ac)) / 2a}$

Example:

Consider the quadratic equation $\mathrm{2x^{2} + 5x - 3 = 0}$ . To solve for x using the quadratic formula:

Step 1: Identify the values of a, b, and c. $\mathrm{a = 2 b = 5 c = -3 }$

Step 2: Substitute the values into the quadratic formula. $\mathrm{x = (-5 \pm \sqrt(5^{2} - 4 \times 2 \times -3)) / (2 \times2) x = (-5 \pm \sqrt(25 + 24)) / 4 x = (-5 \pm \sqrt49) / 4 }$

Step 3: Simplify further if needed. $\mathrm{x_1=(-5+7) / 4=2 / 4=1 / 2 x_2=(-5-7) / 4=-12/4=-3 }$ So, the roots of the quadratic equation $\mathrm{ 2x^{2} + 5x - 3 = 0 \: are\: x = 1/2 \: and \: x = -3. }$

Tips and Tricks:

- Make sure to identify the values of a, b, and c correctly before substituting them into the quadratic formula.

- Remember the $\pm$ symbol in the formula. It indicates that there are two possible solutions.

- To solve the equation using the quadratic formula efficiently, simplify the expression under the square root first to avoid any errors in calculation.

- When the discriminant $\mathrm{(b^{2}- 4ac)}$ is positive, the quadratic equation has two distinct real roots.

- When the discriminant is zero, the quadratic equation has two identical real roots.

- When the discriminant is negative, the quadratic equation has two complex roots. Using the quadratic formula will enable you to solve any quadratic equation accurately. Practice solving various types of quadratic equations to become proficient in using the quadratic formula.

Remember to refer to previous year's management entrance questions to understand the application of the concept in the exams.