Solve 4(x+5)+x^2=900

$$4 ( x + 5 ) + x ^ { 2 } = 900$$

Solve for x

$$x=2\sqrt{221}-2\approx 27.732137495$$

$$x=-2\sqrt{221}-2\approx -31.732137495$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4 by x+5.

$$4x+20+x^{2}=900$$

Subtract 900 from both sides.

$$4x+20+x^{2}-900=0$$

Subtract 900 from 20 to get -880.

$$4x-880+x^{2}=0$$

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.

$$x^{2}+4x-880=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and -880 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-880\right)}}{2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-880\right)}}{2}$$

Multiply -4 times -880.

$$x=\frac{-4±\sqrt{16+3520}}{2}$$

Add 16 to 3520.

$$x=\frac{-4±\sqrt{3536}}{2}$$

Take the square root of 3536.

$$x=\frac{-4±4\sqrt{221}}{2}$$

Now solve the equation x=\frac{-4±4\sqrt{221}}{2} when ± is plus. Add -4 to 4\sqrt{221}.

$$x=\frac{4\sqrt{221}-4}{2}$$

Divide -4+4\sqrt{221} by 2.

$$x=2\sqrt{221}-2$$

Now solve the equation x=\frac{-4±4\sqrt{221}}{2} when ± is minus. Subtract 4\sqrt{221} from -4.

$$x=\frac{-4\sqrt{221}-4}{2}$$

Divide -4-4\sqrt{221} by 2.

$$x=-2\sqrt{221}-2$$

The equation is now solved.

$$x=2\sqrt{221}-2 x=-2\sqrt{221}-2$$

Steps for Completing the Square

Use the distributive property to multiply 4 by x+5.

$$4x+20+x^{2}=900$$

Subtract 20 from both sides.

$$4x+x^{2}=900-20$$

Subtract 20 from 900 to get 880.

$$4x+x^{2}=880$$

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.

$$x^{2}+4x=880$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+4x+2^{2}=880+2^{2}$$

Square 2.

$$x^{2}+4x+4=880+4$$

Add 880 to 4.

$$x^{2}+4x+4=884$$

Factor x^{2}+4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+2\right)^{2}=884$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{884}$$

Simplify.

$$x+2=2\sqrt{221} x+2=-2\sqrt{221}$$

Subtract 2 from both sides of the equation.

$$x=2\sqrt{221}-2 x=-2\sqrt{221}-2$$