Solve 4(x-5)^2-21=-5

$$4 ( x - 5 ) ^ { 2 } - 21 = - 5$$

Solve for x

$$x=7$$

$$x=3$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$4\left(x^{2}-10x+25\right)-21=-5$$

Use the distributive property to multiply 4 by x^{2}-10x+25.

$$4x^{2}-40x+100-21=-5$$

Subtract 21 from 100 to get 79.

$$4x^{2}-40x+79=-5$$

Add 5 to both sides.

$$4x^{2}-40x+79+5=0$$

Add 79 and 5 to get 84.

$$4x^{2}-40x+84=0$$

Divide both sides by 4.

$$x^{2}-10x+21=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+21. To find a and b, set up a system to be solved.

$$a+b=-10 ab=1\times 21=21$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 21.

$$-1,-21 -3,-7$$

Calculate the sum for each pair.

$$-1-21=-22 -3-7=-10$$

The solution is the pair that gives sum -10.

$$a=-7 b=-3$$

Rewrite x^{2}-10x+21 as \left(x^{2}-7x\right)+\left(-3x+21\right).

$$\left(x^{2}-7x\right)+\left(-3x+21\right)$$

Factor out x in the first and -3 in the second group.

$$x\left(x-7\right)-3\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(x-3\right)$$

To find equation solutions, solve x-7=0 and x-3=0.

$$x=7 x=3$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$4\left(x^{2}-10x+25\right)-21=-5$$

Use the distributive property to multiply 4 by x^{2}-10x+25.

$$4x^{2}-40x+100-21=-5$$

Subtract 21 from 100 to get 79.

$$4x^{2}-40x+79=-5$$

Add 5 to both sides.

$$4x^{2}-40x+79+5=0$$

Add 79 and 5 to get 84.

$$4x^{2}-40x+84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -40 for b, and 84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 4\times 84}}{2\times 4}$$

Square -40.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\times 4\times 84}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-40\right)±\sqrt{1600-16\times 84}}{2\times 4}$$

Multiply -16 times 84.

$$x=\frac{-\left(-40\right)±\sqrt{1600-1344}}{2\times 4}$$

Add 1600 to -1344.

$$x=\frac{-\left(-40\right)±\sqrt{256}}{2\times 4}$$

Take the square root of 256.

$$x=\frac{-\left(-40\right)±16}{2\times 4}$$

The opposite of -40 is 40.

$$x=\frac{40±16}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{40±16}{8}$$

Now solve the equation x=\frac{40±16}{8} when ± is plus. Add 40 to 16.

$$x=\frac{56}{8}$$

Divide 56 by 8.

$$x=7$$

Now solve the equation x=\frac{40±16}{8} when ± is minus. Subtract 16 from 40.

$$x=\frac{24}{8}$$

Divide 24 by 8.

$$x=3$$

The equation is now solved.

$$x=7 x=3$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$4\left(x^{2}-10x+25\right)-21=-5$$

Use the distributive property to multiply 4 by x^{2}-10x+25.

$$4x^{2}-40x+100-21=-5$$

Subtract 21 from 100 to get 79.

$$4x^{2}-40x+79=-5$$

Subtract 79 from both sides.

$$4x^{2}-40x=-5-79$$

Subtract 79 from -5 to get -84.

$$4x^{2}-40x=-84$$

Divide both sides by 4.

$$\frac{4x^{2}-40x}{4}=-\frac{84}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{40}{4}\right)x=-\frac{84}{4}$$

Divide -40 by 4.

$$x^{2}-10x=-\frac{84}{4}$$

Divide -84 by 4.

$$x^{2}-10x=-21$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=-21+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-21+25$$

Add -21 to 25.

$$x^{2}-10x+25=4$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{4}$$

Simplify.

$$x-5=2 x-5=-2$$

Add 5 to both sides of the equation.

$$x=7 x=3$$