Question

Find the roots of the quadratic equation 2x² - 7x – 85 = 0 by factorization give answer fast argent​

Answer 1

$\large\underline{\sf{Solution-}}$

Given quadratic equation is

$\rm \: {2x}^{2} - 7x - 85 = 0$

On splitting the middle terms, we get

$\rm \: {2x}^{2} - 17x + 10x - 85 = 0$

$\rm \: x(2x - 17) + 5(2x - 17) = 0$

$\rm \: (2x - 17) \: (x + 5) = 0$

$\rm \: 2x - 17 = 0 \: \: \: or \: \: \: x + 5 = 0$

$\rm\implies \:x = \dfrac{17}{2} \: \: \: or \: \: \: x = - 5$

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Basic Concept Used :-

Splitting of middle terms :-

In order to factorize  ax² + bx + c we have to find numbers m and n such that m + n = b and mn = ac.

After finding m and n, we split the middle term i.e bx in the quadratic equation as mx + nx and get the required factors by grouping the terms.

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Additional Information :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

• If Discriminant, D > 0, then roots of the equation are real and unequal.

• If Discriminant, D = 0, then roots of the equation are real and equal.

• If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

• Discriminant, D = b² - 4ac

Answer 2

$\mathbb\red{ \tiny A \scriptsize \: N \small \:S \large \: W \Large \:E \huge \: R}$

Step-by-step explanation:

• Multiply the coefficient of the first term by the constant term.

2×−85=−170

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• Ask: Which two numbers add up to −7 and multiply to −170?

10 and -17

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• Split −7x as the sum of 10x and −17x.

2x²+10x−17x−85=0

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• Factor out common terms in the first two terms, then in the last two terms.

2x(x+5)−17(x+5)=0

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• Factor out the common term x+5.

(x+5)(2x−17)=0

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When x+5=0 or 2x−17=0

Solve each of the 2 equations above.

$\\ \color{indigo}\boxed{ \bigstar\rm \: \: \: x=-5,\frac{17}{2} \: \: \: \: \bigstar }$

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