Question

How do you solve `x^2 - 10x = -24`?

Answer 1

`x = 4,6`

Explanation:

We have,

`color(white)(xxx) x^2 - 10x = -24`

`rArr x^2 - 10x + 24 = cancel(-24) + cancel24` [Add `24` to both sides]

`rArr x^2 -10x + 24 = 0`

`rArr x^2 - (6 + 4)x + 24 = 0` [Well, you can write `10 = 6 + 4`..]

`rArr x^2 - 6x - 4x + 24 =0` [Break it using Distributive Property]

`rArr x(x - 6) - 4(x - 6) = 0` [Group the like terms]

`rArr (x - 6)(x - 4) = 0` [Group again]

Now, Either `x - 6 = 0 rArr x = 6`

Or, `x - 4 = 0 rArr x = 4`

So, We have Two Solutions, `x = 4,6`.

We can use Quadratic Formula too.

According to the Quadratic Formula,

If there is a Quadratic Equation in the form of `ax^2 + bx + c = 0`,

The roots of the equation are `x = (-b +- sqrt(D))/(2a)`, Where `D = b^2 - 4ac`, which is called the Discriminant.

Now, The Equation in the General Form (`ax^2 + bx + c = 0`) is

`x^2 - 10x + 24 = 0`.

So, `D = (-10)^2 - 4 * 24 * 1 = 100 - 96 = 4 gt 0`

As `D gt 0`, we will have two real and distinct roots for the equation.

Now, `x = (-b +- sqrt(D))/(2a) = (-(-10) +- sqrt(4))/(2 * 1) = (10 + 2)/2, (10 - 2)/2 = 12/2, 8/2= 6,4`

So We get the same solution, `x = 4,6`.

Hope this helps.

Answer 2

`x=4" or "x=6`

Explanation:

`"rearrange in "color(blue)"standard form ";ax^2+bx+c=0`

`"add 24 to both sides"`

`x^2-10x+24=0`

`"the factors of + 24 which sum to - 10 are - 6 and - 4"`

`(x-6)(x-4)=0`

`"equate each factor to zero and solve for x"`

`x-4=0rArrx=4`

`x-6=0rArrx=6`