Solve 13*30*left(1+xright)left(1+5xright)+30*15left(1+5xright)left(1+8xright)=78left(1+10xright)

$$13 \times 30 \times \left( 1+x \right) \left( 1+5x \right) +30 \times 15 \left( 1+5x \right) \left( 1+8x \right) = 78 \left( 1+10x \right)$$

Solve for x (complex solution)

$$x=\frac{\sqrt{6555}i}{1330}-\frac{13}{70}\approx -0.185714286+0.060874402i$$

$$x=-\frac{\sqrt{6555}i}{1330}-\frac{13}{70}\approx -0.185714286-0.060874402i$$

Steps Using the Quadratic Formula

Do the multiplications.

$$390\left(1+x\right)\left(1+5x\right)+450\left(1+5x\right)\left(1+8x\right)=78\left(1+10x\right)$$

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

$$\left(390+390x\right)\left(1+5x\right)+450\left(1+5x\right)\left(1+8x\right)=78\left(1+10x\right)$$

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

$$390+2340x+1950x^{2}+450\left(1+5x\right)\left(1+8x\right)=78\left(1+10x\right)$$

Use the distributive property to multiply 450 by 1+5x.

$$390+2340x+1950x^{2}+\left(450+2250x\right)\left(1+8x\right)=78\left(1+10x\right)$$

Use the distributive property to multiply 450+2250x by 1+8x and combine like terms.

$$390+2340x+1950x^{2}+450+5850x+18000x^{2}=78\left(1+10x\right)$$

Add 390 and 450 to get 840.

$$840+2340x+1950x^{2}+5850x+18000x^{2}=78\left(1+10x\right)$$

Combine 2340x and 5850x to get 8190x.

$$840+8190x+1950x^{2}+18000x^{2}=78\left(1+10x\right)$$

Combine 1950x^{2} and 18000x^{2} to get 19950x^{2}.

$$840+8190x+19950x^{2}=78\left(1+10x\right)$$

Use the distributive property to multiply 78 by 1+10x.

$$840+8190x+19950x^{2}=78+780x$$

Subtract 78 from both sides.

$$840+8190x+19950x^{2}-78=780x$$

Subtract 78 from 840 to get 762.

$$762+8190x+19950x^{2}=780x$$

Subtract 780x from both sides.

$$762+8190x+19950x^{2}-780x=0$$

Combine 8190x and -780x to get 7410x.

$$762+7410x+19950x^{2}=0$$

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.

$$19950x^{2}+7410x+762=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 19950 for a, 7410 for b, and 762 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7410±\sqrt{7410^{2}-4\times 19950\times 762}}{2\times 19950}$$

Square 7410.

$$x=\frac{-7410±\sqrt{54908100-4\times 19950\times 762}}{2\times 19950}$$

Multiply -4 times 19950.

$$x=\frac{-7410±\sqrt{54908100-79800\times 762}}{2\times 19950}$$

Multiply -79800 times 762.

$$x=\frac{-7410±\sqrt{54908100-60807600}}{2\times 19950}$$

Add 54908100 to -60807600.

$$x=\frac{-7410±\sqrt{-5899500}}{2\times 19950}$$

Take the square root of -5899500.

$$x=\frac{-7410±30\sqrt{6555}i}{2\times 19950}$$

Multiply 2 times 19950.

$$x=\frac{-7410±30\sqrt{6555}i}{39900}$$

Now solve the equation x=\frac{-7410±30\sqrt{6555}i}{39900} when ± is plus. Add -7410 to 30i\sqrt{6555}.

$$x=\frac{-7410+30\sqrt{6555}i}{39900}$$

Divide -7410+30i\sqrt{6555} by 39900.

$$x=\frac{\sqrt{6555}i}{1330}-\frac{13}{70}$$

Now solve the equation x=\frac{-7410±30\sqrt{6555}i}{39900} when ± is minus. Subtract 30i\sqrt{6555} from -7410.

$$x=\frac{-30\sqrt{6555}i-7410}{39900}$$

Divide -7410-30i\sqrt{6555} by 39900.

$$x=-\frac{\sqrt{6555}i}{1330}-\frac{13}{70}$$

The equation is now solved.

$$x=\frac{\sqrt{6555}i}{1330}-\frac{13}{70} x=-\frac{\sqrt{6555}i}{1330}-\frac{13}{70}$$

Steps for Completing the Square

Do the multiplications.

$$390\left(1+x\right)\left(1+5x\right)+450\left(1+5x\right)\left(1+8x\right)=78\left(1+10x\right)$$

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

$$\left(390+390x\right)\left(1+5x\right)+450\left(1+5x\right)\left(1+8x\right)=78\left(1+10x\right)$$

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

$$390+2340x+1950x^{2}+450\left(1+5x\right)\left(1+8x\right)=78\left(1+10x\right)$$

Use the distributive property to multiply 450 by 1+5x.

$$390+2340x+1950x^{2}+\left(450+2250x\right)\left(1+8x\right)=78\left(1+10x\right)$$

Use the distributive property to multiply 450+2250x by 1+8x and combine like terms.

$$390+2340x+1950x^{2}+450+5850x+18000x^{2}=78\left(1+10x\right)$$

Add 390 and 450 to get 840.

$$840+2340x+1950x^{2}+5850x+18000x^{2}=78\left(1+10x\right)$$

Combine 2340x and 5850x to get 8190x.

$$840+8190x+1950x^{2}+18000x^{2}=78\left(1+10x\right)$$

Combine 1950x^{2} and 18000x^{2} to get 19950x^{2}.

$$840+8190x+19950x^{2}=78\left(1+10x\right)$$

Use the distributive property to multiply 78 by 1+10x.

$$840+8190x+19950x^{2}=78+780x$$

Subtract 780x from both sides.

$$840+8190x+19950x^{2}-780x=78$$

Combine 8190x and -780x to get 7410x.

$$840+7410x+19950x^{2}=78$$

Subtract 840 from both sides.

$$7410x+19950x^{2}=78-840$$

Subtract 840 from 78 to get -762.

$$7410x+19950x^{2}=-762$$

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.

$$19950x^{2}+7410x=-762$$

Divide both sides by 19950.

$$\frac{19950x^{2}+7410x}{19950}=-\frac{762}{19950}$$

Dividing by 19950 undoes the multiplication by 19950.

$$x^{2}+\frac{7410}{19950}x=-\frac{762}{19950}$$

Reduce the fraction \frac{7410}{19950} to lowest terms by extracting and canceling out 570.

$$x^{2}+\frac{13}{35}x=-\frac{762}{19950}$$

Reduce the fraction \frac{-762}{19950} to lowest terms by extracting and canceling out 6.

$$x^{2}+\frac{13}{35}x=-\frac{127}{3325}$$

Divide \frac{13}{35}, the coefficient of the x term, by 2 to get \frac{13}{70}. Then add the square of \frac{13}{70} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{13}{35}x+\left(\frac{13}{70}\right)^{2}=-\frac{127}{3325}+\left(\frac{13}{70}\right)^{2}$$

Square \frac{13}{70} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{13}{35}x+\frac{169}{4900}=-\frac{127}{3325}+\frac{169}{4900}$$

Add -\frac{127}{3325} to \frac{169}{4900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{13}{35}x+\frac{169}{4900}=-\frac{69}{18620}$$

Factor x^{2}+\frac{13}{35}x+\frac{169}{4900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{13}{70}\right)^{2}=-\frac{69}{18620}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{70}\right)^{2}}=\sqrt{-\frac{69}{18620}}$$

Simplify.

$$x+\frac{13}{70}=\frac{\sqrt{6555}i}{1330} x+\frac{13}{70}=-\frac{\sqrt{6555}i}{1330}$$

Subtract \frac{13}{70} from both sides of the equation.

$$x=\frac{\sqrt{6555}i}{1330}-\frac{13}{70} x=-\frac{\sqrt{6555}i}{1330}-\frac{13}{70}$$