Question

X²-5x-14=0 find the roots of qudratic equation​

Answer 1

Answer:The first term is, x2 its coefficient is 1 .

The middle term is, -5x its coefficient is -5 .

The last term, "the constant", is -14

Step-1 : Multiply the coefficient of the first term by the constant 1 • -14 = -14

Step-2 : Find two factors of -14 whose sum equals the coefficient of the middle term, which is -5 .

-14 + 1 = -13

-7 + 2 = -5 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -7 and 2

x2 - 7x + 2x - 14

Step-4 : Add up the first 2 terms, pulling out like factors :

x • (x-7)

Add up the last 2 terms, pulling out common factors :

2 • (x-7)

Step-5 : Add up the four terms of step 4 :

(x+2) • (x-7)

Which is the desired factorization

Equation at the end of step

1

:

(x + 2) • (x - 7) = 0

STEP

2

:

Theory - Roots of a product

2.1 A product of several terms equals zero.

When a product of two or more terms equals zero, then at least one of the terms must be zero.

We shall now solve each term = 0 separately

In other words, we are going to solve as many equations as there are terms in the product

Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation:

2.2 Solve : x+2 = 0

Subtract 2 from both sides of the equation :

x = -2

Solving a Single Variable Equation:

2.3 Solve : x-7 = 0

Add 7 to both sides of the equation :

x = 7

Supplement : Solving Quadratic Equation Directly

Solving x2-5x-14 = 0 directly

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Step-by-step explanation:

Answer 2

Explanation:—

Using quadratic formula :–

$\dfrac{–b \: ±\:\sqrt{ b{}^{2 \: } - 4 \: \: (ac)}}{2a}$

__________________

Substituting the value a = 1, b = -5 and c = -14

into the quadratic formula

.

Solving for x :

$\dfrac{5 \: ±\:\sqrt{( - 5) {}^{2 \: } - 4 \: × \: (1 × - 14)}}{2×1}$

$x \: = \: \dfrac{5 ± \sqrt{25 - 4\:× - 14} }{2×1}$

$x \: = \: \dfrac{5 \: ± \: \sqrt{25 \: + \: 56} }{2×1}$

$x \: = \: \dfrac{5 \: ± \: \sqrt{81} }{2×1}$

$x \: = \: \dfrac{5 \: ± \: \sqrt{9 {}^{2} } }{2×1}$

$x \: = \: \dfrac{5 \: ±\: 9}{2}$

$x\:=\: 7,\: -2$

______________________

Final answer:—

$x\:=\: 7,\: -2$