Solve 10x+3x^2+(10+6x)^2/10=115

$$10 x + 3 x ^ { 2 } + \frac { ( 10 + 6 x ) ^ { 2 } } { 10 } = 115$$

Solve for x

$$x=\frac{5\sqrt{814}}{33}-\frac{5}{3}\approx 2.65616443$$

$$x=-\frac{5\sqrt{814}}{33}-\frac{5}{3}\approx -5.989497763$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 10.

$$100x+30x^{2}+\left(10+6x\right)^{2}=1150$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(10+6x\right)^{2}.

$$100x+30x^{2}+100+120x+36x^{2}=1150$$

Combine 100x and 120x to get 220x.

$$220x+30x^{2}+100+36x^{2}=1150$$

Combine 30x^{2} and 36x^{2} to get 66x^{2}.

$$220x+66x^{2}+100=1150$$

Subtract 1150 from both sides.

$$220x+66x^{2}+100-1150=0$$

Subtract 1150 from 100 to get -1050.

$$220x+66x^{2}-1050=0$$

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.

$$66x^{2}+220x-1050=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 66 for a, 220 for b, and -1050 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-220±\sqrt{220^{2}-4\times 66\left(-1050\right)}}{2\times 66}$$

Square 220.

$$x=\frac{-220±\sqrt{48400-4\times 66\left(-1050\right)}}{2\times 66}$$

Multiply -4 times 66.

$$x=\frac{-220±\sqrt{48400-264\left(-1050\right)}}{2\times 66}$$

Multiply -264 times -1050.

$$x=\frac{-220±\sqrt{48400+277200}}{2\times 66}$$

Add 48400 to 277200.

$$x=\frac{-220±\sqrt{325600}}{2\times 66}$$

Take the square root of 325600.

$$x=\frac{-220±20\sqrt{814}}{2\times 66}$$

Multiply 2 times 66.

$$x=\frac{-220±20\sqrt{814}}{132}$$

Now solve the equation x=\frac{-220±20\sqrt{814}}{132} when ± is plus. Add -220 to 20\sqrt{814}.

$$x=\frac{20\sqrt{814}-220}{132}$$

Divide -220+20\sqrt{814} by 132.

$$x=\frac{5\sqrt{814}}{33}-\frac{5}{3}$$

Now solve the equation x=\frac{-220±20\sqrt{814}}{132} when ± is minus. Subtract 20\sqrt{814} from -220.

$$x=\frac{-20\sqrt{814}-220}{132}$$

Divide -220-20\sqrt{814} by 132.

$$x=-\frac{5\sqrt{814}}{33}-\frac{5}{3}$$

The equation is now solved.

$$x=\frac{5\sqrt{814}}{33}-\frac{5}{3} x=-\frac{5\sqrt{814}}{33}-\frac{5}{3}$$

Steps for Completing the Square

Multiply both sides of the equation by 10.

$$100x+30x^{2}+\left(10+6x\right)^{2}=1150$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(10+6x\right)^{2}.

$$100x+30x^{2}+100+120x+36x^{2}=1150$$

Combine 100x and 120x to get 220x.

$$220x+30x^{2}+100+36x^{2}=1150$$

Combine 30x^{2} and 36x^{2} to get 66x^{2}.

$$220x+66x^{2}+100=1150$$

Subtract 100 from both sides.

$$220x+66x^{2}=1150-100$$

Subtract 100 from 1150 to get 1050.

$$220x+66x^{2}=1050$$

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.

$$66x^{2}+220x=1050$$

Divide both sides by 66.

$$\frac{66x^{2}+220x}{66}=\frac{1050}{66}$$

Dividing by 66 undoes the multiplication by 66.

$$x^{2}+\frac{220}{66}x=\frac{1050}{66}$$

Reduce the fraction \frac{220}{66} to lowest terms by extracting and canceling out 22.

$$x^{2}+\frac{10}{3}x=\frac{1050}{66}$$

Reduce the fraction \frac{1050}{66} to lowest terms by extracting and canceling out 6.

$$x^{2}+\frac{10}{3}x=\frac{175}{11}$$

Divide \frac{10}{3}, the coefficient of the x term, by 2 to get \frac{5}{3}. Then add the square of \frac{5}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{10}{3}x+\left(\frac{5}{3}\right)^{2}=\frac{175}{11}+\left(\frac{5}{3}\right)^{2}$$

Square \frac{5}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{10}{3}x+\frac{25}{9}=\frac{175}{11}+\frac{25}{9}$$

Add \frac{175}{11} to \frac{25}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{10}{3}x+\frac{25}{9}=\frac{1850}{99}$$

Factor x^{2}+\frac{10}{3}x+\frac{25}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{3}\right)^{2}=\frac{1850}{99}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{3}\right)^{2}}=\sqrt{\frac{1850}{99}}$$

Simplify.

$$x+\frac{5}{3}=\frac{5\sqrt{814}}{33} x+\frac{5}{3}=-\frac{5\sqrt{814}}{33}$$

Subtract \frac{5}{3} from both sides of the equation.

$$x=\frac{5\sqrt{814}}{33}-\frac{5}{3} x=-\frac{5\sqrt{814}}{33}-\frac{5}{3}$$