Solve 13*30*left(1+xright)left(1+5xright)+26*15left(1+5xright)left(1+8xright)=780left(1+10xright)

$$13 \times 30 \times \left( 1+x \right) \left( 1+5x \right) +26 \times 15 \left( 1+5x \right) \left( 1+8x \right) = 780 \left( 1+10x \right)$$

Solve for x

$$x=\frac{1}{45}\approx 0.022222222$$

$$x=0$$

Steps Using the Quadratic Formula

Do the multiplications.

$$390\left(1+x\right)\left(1+5x\right)+390\left(1+5x\right)\left(1+8x\right)=780\left(1+10x\right)$$

Use the distributive property to multiply 390 by 1+x.

$$\left(390+390x\right)\left(1+5x\right)+390\left(1+5x\right)\left(1+8x\right)=780\left(1+10x\right)$$

Use the distributive property to multiply 390+390x by 1+5x and combine like terms.

$$390+2340x+1950x^{2}+390\left(1+5x\right)\left(1+8x\right)=780\left(1+10x\right)$$

Use the distributive property to multiply 390 by 1+5x.

$$390+2340x+1950x^{2}+\left(390+1950x\right)\left(1+8x\right)=780\left(1+10x\right)$$

Use the distributive property to multiply 390+1950x by 1+8x and combine like terms.

$$390+2340x+1950x^{2}+390+5070x+15600x^{2}=780\left(1+10x\right)$$

Add 390 and 390 to get 780.

$$780+2340x+1950x^{2}+5070x+15600x^{2}=780\left(1+10x\right)$$

Combine 2340x and 5070x to get 7410x.

$$780+7410x+1950x^{2}+15600x^{2}=780\left(1+10x\right)$$

Combine 1950x^{2} and 15600x^{2} to get 17550x^{2}.

$$780+7410x+17550x^{2}=780\left(1+10x\right)$$

Use the distributive property to multiply 780 by 1+10x.

$$780+7410x+17550x^{2}=780+7800x$$

Subtract 780 from both sides.

$$780+7410x+17550x^{2}-780=7800x$$

Subtract 780 from 780 to get 0.

$$7410x+17550x^{2}=7800x$$

Subtract 7800x from both sides.

$$7410x+17550x^{2}-7800x=0$$

Combine 7410x and -7800x to get -390x.

$$-390x+17550x^{2}=0$$

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.

$$17550x^{2}-390x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 17550 for a, -390 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-390\right)±\sqrt{\left(-390\right)^{2}}}{2\times 17550}$$

Take the square root of \left(-390\right)^{2}.

$$x=\frac{-\left(-390\right)±390}{2\times 17550}$$

The opposite of -390 is 390.

$$x=\frac{390±390}{2\times 17550}$$

Multiply 2 times 17550.

$$x=\frac{390±390}{35100}$$

Now solve the equation x=\frac{390±390}{35100} when ± is plus. Add 390 to 390.

$$x=\frac{780}{35100}$$

Reduce the fraction \frac{780}{35100} to lowest terms by extracting and canceling out 780.

$$x=\frac{1}{45}$$

Now solve the equation x=\frac{390±390}{35100} when ± is minus. Subtract 390 from 390.

$$x=\frac{0}{35100}$$

Divide 0 by 35100.

$$x=0$$

The equation is now solved.

$$x=\frac{1}{45} x=0$$

Steps for Completing the Square

Do the multiplications.

$$390\left(1+x\right)\left(1+5x\right)+390\left(1+5x\right)\left(1+8x\right)=780\left(1+10x\right)$$

Use the distributive property to multiply 390 by 1+x.

$$\left(390+390x\right)\left(1+5x\right)+390\left(1+5x\right)\left(1+8x\right)=780\left(1+10x\right)$$

Use the distributive property to multiply 390+390x by 1+5x and combine like terms.

$$390+2340x+1950x^{2}+390\left(1+5x\right)\left(1+8x\right)=780\left(1+10x\right)$$

Use the distributive property to multiply 390 by 1+5x.

$$390+2340x+1950x^{2}+\left(390+1950x\right)\left(1+8x\right)=780\left(1+10x\right)$$

Use the distributive property to multiply 390+1950x by 1+8x and combine like terms.

$$390+2340x+1950x^{2}+390+5070x+15600x^{2}=780\left(1+10x\right)$$

Add 390 and 390 to get 780.

$$780+2340x+1950x^{2}+5070x+15600x^{2}=780\left(1+10x\right)$$

Combine 2340x and 5070x to get 7410x.

$$780+7410x+1950x^{2}+15600x^{2}=780\left(1+10x\right)$$

Combine 1950x^{2} and 15600x^{2} to get 17550x^{2}.

$$780+7410x+17550x^{2}=780\left(1+10x\right)$$

Use the distributive property to multiply 780 by 1+10x.

$$780+7410x+17550x^{2}=780+7800x$$

Subtract 7800x from both sides.

$$780+7410x+17550x^{2}-7800x=780$$

Combine 7410x and -7800x to get -390x.

$$780-390x+17550x^{2}=780$$

Subtract 780 from both sides.

$$-390x+17550x^{2}=780-780$$

Subtract 780 from 780 to get 0.

$$-390x+17550x^{2}=0$$

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.

$$17550x^{2}-390x=0$$

Divide both sides by 17550.

$$\frac{17550x^{2}-390x}{17550}=\frac{0}{17550}$$

Dividing by 17550 undoes the multiplication by 17550.

$$x^{2}+\left(-\frac{390}{17550}\right)x=\frac{0}{17550}$$

Reduce the fraction \frac{-390}{17550} to lowest terms by extracting and canceling out 390.

$$x^{2}-\frac{1}{45}x=\frac{0}{17550}$$

Divide 0 by 17550.

$$x^{2}-\frac{1}{45}x=0$$

Divide -\frac{1}{45}, the coefficient of the x term, by 2 to get -\frac{1}{90}. Then add the square of -\frac{1}{90} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{45}x+\left(-\frac{1}{90}\right)^{2}=\left(-\frac{1}{90}\right)^{2}$$

Square -\frac{1}{90} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{45}x+\frac{1}{8100}=\frac{1}{8100}$$

Factor x^{2}-\frac{1}{45}x+\frac{1}{8100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{90}\right)^{2}=\frac{1}{8100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{90}\right)^{2}}=\sqrt{\frac{1}{8100}}$$

Simplify.

$$x-\frac{1}{90}=\frac{1}{90} x-\frac{1}{90}=-\frac{1}{90}$$

Add \frac{1}{90} to both sides of the equation.

$$x=\frac{1}{45} x=0$$