Question

How do you factor and solve ${x^2} - 4x - 21 = 0?$

Answer 1

Hint:We can find the factors of a quadratic equation $\left( {a{x^2} + bx + c = 0} \right)$ by the formula $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$. The quadratic equations have two factors. we can find the factors either by the above formula or by factoring method where we split the middle term and by taking common, we can find the respective factors.

Complete step by step answer:
We are given a quadratic equation ${x^2} - 4x - 21 = 0$. Finding factors by splitting the middle term, in this method we have to find two numbers whose addition is the coefficient of x here is -$4$ and multiplication is the constant part here is -$21$. Now, these two numbers are $ - 7,3$ which satisfies the above two conditions. Hence, we are splitting the middle term,
${x^2} - 7x + 3x - 21 = 0$
Now, taking common from the first two terms and the last two terms respectively,
$x\left( {x - 7} \right) + 3\left( {x - 7} \right) = 0$
Now, $\left( {x - 7} \right)\left( {x + 3} \right) = 0$
Equating the two brackets to zero. Hence, the factors of the given equation are $7, - 3$

Note:You have to be very careful while selecting the two numbers. Signs should also be taken care of while selecting the numbers. The sum of the selected numbers is the coefficient of x and the product of numbers is the constant term. If we are unable to find the numbers use the formula above written in the hint part.