Solve (30,1^2-2^2)*(15^2-30a+a^2)=750

$$( 30,1 ^ { 2 } - 2 ^ { 2 } ) \cdot ( 15 ^ { 2 } - 30 a + a ^ { 2 } ) = 750$$

Solve for a

$$a=\frac{5\sqrt{195}}{13}+15\approx 20.370861555$$

$$a=-\frac{5\sqrt{195}}{13}+15\approx 9.629138445$$

Steps Using the Quadratic Formula

Calculate 1 to the power of 2 and get 1.

$$\left(30\times 1-2^{2}\right)\left(15^{2}-30a+a^{2}\right)=750$$

Multiply 30 and 1 to get 30.

$$\left(30-2^{2}\right)\left(15^{2}-30a+a^{2}\right)=750$$

Calculate 2 to the power of 2 and get 4.

$$\left(30-4\right)\left(15^{2}-30a+a^{2}\right)=750$$

Subtract 4 from 30 to get 26.

$$26\left(15^{2}-30a+a^{2}\right)=750$$

Calculate 15 to the power of 2 and get 225.

$$26\left(225-30a+a^{2}\right)=750$$

Use the distributive property to multiply 26 by 225-30a+a^{2}.

$$5850-780a+26a^{2}=750$$

Subtract 750 from both sides.

$$5850-780a+26a^{2}-750=0$$

Subtract 750 from 5850 to get 5100.

$$5100-780a+26a^{2}=0$$

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.

$$26a^{2}-780a+5100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 26 for a, -780 for b, and 5100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-780\right)±\sqrt{\left(-780\right)^{2}-4\times 26\times 5100}}{2\times 26}$$

Square -780.

$$a=\frac{-\left(-780\right)±\sqrt{608400-4\times 26\times 5100}}{2\times 26}$$

Multiply -4 times 26.

$$a=\frac{-\left(-780\right)±\sqrt{608400-104\times 5100}}{2\times 26}$$

Multiply -104 times 5100.

$$a=\frac{-\left(-780\right)±\sqrt{608400-530400}}{2\times 26}$$

Add 608400 to -530400.

$$a=\frac{-\left(-780\right)±\sqrt{78000}}{2\times 26}$$

Take the square root of 78000.

$$a=\frac{-\left(-780\right)±20\sqrt{195}}{2\times 26}$$

The opposite of -780 is 780.

$$a=\frac{780±20\sqrt{195}}{2\times 26}$$

Multiply 2 times 26.

$$a=\frac{780±20\sqrt{195}}{52}$$

Now solve the equation a=\frac{780±20\sqrt{195}}{52} when ± is plus. Add 780 to 20\sqrt{195}.

$$a=\frac{20\sqrt{195}+780}{52}$$

Divide 780+20\sqrt{195} by 52.

$$a=\frac{5\sqrt{195}}{13}+15$$

Now solve the equation a=\frac{780±20\sqrt{195}}{52} when ± is minus. Subtract 20\sqrt{195} from 780.

$$a=\frac{780-20\sqrt{195}}{52}$$

Divide 780-20\sqrt{195} by 52.

$$a=-\frac{5\sqrt{195}}{13}+15$$

The equation is now solved.

$$a=\frac{5\sqrt{195}}{13}+15 a=-\frac{5\sqrt{195}}{13}+15$$

Steps for Completing the Square

Calculate 1 to the power of 2 and get 1.

$$\left(30\times 1-2^{2}\right)\left(15^{2}-30a+a^{2}\right)=750$$

Multiply 30 and 1 to get 30.

$$\left(30-2^{2}\right)\left(15^{2}-30a+a^{2}\right)=750$$

Calculate 2 to the power of 2 and get 4.

$$\left(30-4\right)\left(15^{2}-30a+a^{2}\right)=750$$

Subtract 4 from 30 to get 26.

$$26\left(15^{2}-30a+a^{2}\right)=750$$

Calculate 15 to the power of 2 and get 225.

$$26\left(225-30a+a^{2}\right)=750$$

Use the distributive property to multiply 26 by 225-30a+a^{2}.

$$5850-780a+26a^{2}=750$$

Subtract 5850 from both sides.

$$-780a+26a^{2}=750-5850$$

Subtract 5850 from 750 to get -5100.

$$-780a+26a^{2}=-5100$$

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.

$$26a^{2}-780a=-5100$$

Divide both sides by 26.

$$\frac{26a^{2}-780a}{26}=-\frac{5100}{26}$$

Dividing by 26 undoes the multiplication by 26.

$$a^{2}+\left(-\frac{780}{26}\right)a=-\frac{5100}{26}$$

Divide -780 by 26.

$$a^{2}-30a=-\frac{5100}{26}$$

Reduce the fraction \frac{-5100}{26} to lowest terms by extracting and canceling out 2.

$$a^{2}-30a=-\frac{2550}{13}$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-30a+\left(-15\right)^{2}=-\frac{2550}{13}+\left(-15\right)^{2}$$

Square -15.

$$a^{2}-30a+225=-\frac{2550}{13}+225$$

Add -\frac{2550}{13} to 225.

$$a^{2}-30a+225=\frac{375}{13}$$

Factor a^{2}-30a+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-15\right)^{2}=\frac{375}{13}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-15\right)^{2}}=\sqrt{\frac{375}{13}}$$

Simplify.

$$a-15=\frac{5\sqrt{195}}{13} a-15=-\frac{5\sqrt{195}}{13}$$

Add 15 to both sides of the equation.

$$a=\frac{5\sqrt{195}}{13}+15 a=-\frac{5\sqrt{195}}{13}+15$$