Question

For what values of q does the quadratic equation
x^2+2(q_4)x+2q=0

Answer 1

${\large \green{\underline{\underline{Answer}}}} \\ \bold{values \: of \: q = 2 \: \: and \: \: q = 8}$

$\mathfrak{\large \red{\underline{\underline{given}}}} \\ \large \bold{ {x}^{2} + 2(q - 4)x + 2q = 0 } \\ \mathfrak{\large \green{\underline{\underline{to \: find \: the \: values \: of \: q \: }}}} \\ \mathfrak{\large \red {\underline{\underline{explanation}}}} \\ \\ \large \bold{ \red{ \underline{ \underline{formula}}}} \\ \large \bold{ \green{d = {b}^{2} - 4ac = 0 }} \\ \bold{2(q - 4) {}^{2} - 4(1)( + 2q) = 0 } \\ \bold{4( {q}^{2} + 16 - 8q) - 8q = 0 } \\ \bold{4 {q}^{2} + 64 - 32q - 8q = 0 } \\ \bold{4 {q}^{2} - 40q + 64 = 0 } \\ \bold{4( {q}^{2} - 10q + 16) = 0 } \\ \bold{ {q}^{2} - 10q + 16 = 0 } \\ \\ {\large \red{\underline{\underline{factorise \: into \: middle \: term \: split}}}} \\ \bold{ {q}^{2} - 8q - 2q + 16 = 0 } \\ \bold{q(q - 8) - 2(q - 8) = 0} \\ \bold{(q - 2)(q - 8) = 0} \\ \large \green{q = 2 \: and \: \: q = 8}$

Answer 2

QUESTION:

For what values of q does the quadratic equation

x^2+2(q_4)x+2q=0

CONCEPT USED :

We know that; quadratic polynomial is in the form of

$\huge\orange {a {x}^{2} + bx + c = 0}$

where;

b = coefficient of x

a = coefficient of x square

c = constant term

we use the discriminant (d) formula to find the value of q.

$\huge\blue {d = {b}^{2} - 4ac}$

now come to main question;

Here;

a = + 1

b = +2(q-4) = +2q-8

c = +2q

using the formula;

$d = {b}^{2} - 4ac \\ 0 = ({2q - 8})^{2} - 4 \times 1 \times 2q$

$0 = {(2q)}^{2} - 2 \times 2q \times8 + {8}^{2} - 8q \\ 0 = 4 {q}^{2} - 32q - 8q + 64 \\ 0 = 4 {q}^{2} - 40q + 64$

$4( {q}^{2} - 10q + 16) = 0$

$\red {(taking \: 4 \: common)}$

${q}^{2} - 10q + 16 = 0$

now it is a quadratic equation we have to factorised it,

Splitting the middle term in such a way that it's sum equal to -10 and product equal to 16.

so,

${q}^{2} - 8q - 2q + 16 = 0 \\ q(q - 8) - 2(q - 8) = 0 \\ (q - 8)(q - 2) = 0$

hence;

$</strong><strong>\</strong><strong>huge</strong><strong>\</strong><strong>green</strong><strong> </strong><strong>{</strong><strong>(q - 8)(q - 2) = 0</strong><strong>}</strong><strong>$

so,

$q - 8 = 0 \\ q = 8 \\ \\ or \ \\\\q - 2 = 0 \\ q = 2$

FINAL ANSWER :

$</strong><strong>\</strong><strong>h</strong><strong>u</strong><strong>g</strong><strong>e</strong><strong>\</strong><strong>o</strong><strong>r</strong><strong>a</strong><strong>n</strong><strong>g</strong><strong>e</strong><strong> </strong><strong>{</strong><strong>q = 2 \: or \: 8</strong><strong>}</strong><strong>$