Solve (5+x)^2+4.8^2=8^2

$$( 5 + x ) ^ { 2 } + 4.8 ^ { 2 } = 8 ^ { 2 }$$

Solve for x

$$x=1.4$$

$$x=-11.4$$

Steps Using the Quadratic Formula

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

$$25+10x+x^{2}+4.8^{2}=8^{2}$$

Calculate 4.8 to the power of 2 and get 23.04.

$$25+10x+x^{2}+23.04=8^{2}$$

Add 25 and 23.04 to get 48.04.

$$48.04+10x+x^{2}=8^{2}$$

Calculate 8 to the power of 2 and get 64.

$$48.04+10x+x^{2}=64$$

Subtract 64 from both sides.

$$48.04+10x+x^{2}-64=0$$

Subtract 64 from 48.04 to get -15.96.

$$-15.96+10x+x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+10x-15.96=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 10 for b, and -15.96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-15.96\right)}}{2}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-15.96\right)}}{2}$$

Multiply -4 times -15.96.

$$x=\frac{-10±\sqrt{100+63.84}}{2}$$

Add 100 to 63.84.

$$x=\frac{-10±\sqrt{163.84}}{2}$$

Take the square root of 163.84.

$$x=\frac{-10±\frac{64}{5}}{2}$$

Now solve the equation x=\frac{-10±\frac{64}{5}}{2} when ± is plus. Add -10 to \frac{64}{5}.

$$x=\frac{\frac{14}{5}}{2}$$

Divide \frac{14}{5} by 2.

$$x=\frac{7}{5}$$

Now solve the equation x=\frac{-10±\frac{64}{5}}{2} when ± is minus. Subtract \frac{64}{5} from -10.

$$x=-\frac{\frac{114}{5}}{2}$$

Divide -\frac{114}{5} by 2.

$$x=-\frac{57}{5}$$

The equation is now solved.

$$x=\frac{7}{5} x=-\frac{57}{5}$$

Steps for Completing the Square

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

$$25+10x+x^{2}+4.8^{2}=8^{2}$$

Calculate 4.8 to the power of 2 and get 23.04.

$$25+10x+x^{2}+23.04=8^{2}$$

Add 25 and 23.04 to get 48.04.

$$48.04+10x+x^{2}=8^{2}$$

Calculate 8 to the power of 2 and get 64.

$$48.04+10x+x^{2}=64$$

Subtract 48.04 from both sides.

$$10x+x^{2}=64-48.04$$

Subtract 48.04 from 64 to get 15.96.

$$10x+x^{2}=15.96$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+10x=15.96$$

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.

$$x^{2}+10x+5^{2}=15.96+5^{2}$$

Square 5.

$$x^{2}+10x+25=15.96+25$$

Add 15.96 to 25.

$$x^{2}+10x+25=40.96$$

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}.

$$\left(x+5\right)^{2}=40.96$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{40.96}$$

Simplify.

$$x+5=\frac{32}{5} x+5=-\frac{32}{5}$$

Subtract 5 from both sides of the equation.

$$x=\frac{7}{5} x=-\frac{57}{5}$$