Question

How do you solve ${k^2} = 76$?

Answer 1

In this question we have to solve the quadratic equation ${k^2} = 76$.Given equation is quadratic equation because it is of the form $a{x^2} + bx + c = 0$.Given a quadratic equation doesn’t have a term of the form $bx$. To solve this we need to know the quadratic formula and discriminant of a quadratic equation. Discriminant of a quadratic equation gives details about the nature of the roots of a quadratic equation.

Complete step by step answer:
Let us try to solve this question in which we are asked to solve for ${k^2} + 0k - 76 = 0$ equation ${k^2} = 76$. This equation can be written as ${k^2} + 0k - 76 = 0$. To solve this quadratic equation we will use quadratic formula and it is given by $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ for any general quadratic equation $a{x^2} + bx + c = 0$ where ${b^2} - 4ac$ is called the discriminant of quadratic equation, it tells the nature of roots of quadratic equation. Here are conditions:
-Two distinct real roots, if ${b^2} - 4ac > 0$
-Two equal real roots, if ${b^2} - 4ac = 0$
-No real roots, if ${b^2} - 4ac < 0$
In the given quadratic equation ${k^2} = 76$ we have,
a = 1 \\\ \Rightarrow b = 0 \\\ \Rightarrow c = - 76 \\\
Discriminant of the quadratic equation is
{b^2} - 4ac = {(0)^2} - 4 \cdot 1 \cdot ( - 76) \\\ \Rightarrow{b^2} - 4ac = 304 > 0 \\\
Hence the given quadratic equation has two distinct real roots, because discriminant is greater than $0$.Now putting values of $a,b$and $c$in quadratic formula we get,

$$k = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} \\\ \Rightarrow k= \dfrac{{ - (0) \pm \sqrt {{{(0)}^2} - 4 \cdot 1 \cdot ( - 76)} }}{2} \\\ \Rightarrow k= \dfrac{{0 \pm \sqrt {304} }}{2} \\\ \Rightarrow k = \dfrac{{ \pm \sqrt {304} }}{2} \\\$$

\Rightarrow k = \dfrac{{ \pm \sqrt {304} }}{2} \\\ \Rightarrow k= \dfrac{{ \pm \sqrt {16 \times 19} }}{2} \\\ \therefore k = \dfrac{{ \pm 4\sqrt {19} }}{2} \\\
Because we know that $\sqrt {16} = \pm 4$. As we know $4$ multiple of $2$.

Hence the value of $k = \pm 2\sqrt {19}$.

Note: To solve questions in which you are asked to find the roots of quadratic equations by quadratic formula you must need to know the formula. We can also solve this by using other methods of finding roots of a quadratic equation such as completing square method and factor method.