Solve (25)^2+(4-a)^2=1.5a^2

$$( 25 ) ^ { 2 } + ( 4 - a ) ^ { 2 } = 1.5 a ^ { 2 }$$

Solve for a (complex solution)

$$a=\sqrt{1346}-8\approx 28.687872656$$

$$a=-\left(\sqrt{1346}+8\right)\approx -44.687872656$$

Steps Using the Quadratic Formula

Calculate 25 to the power of 2 and get 625.

$$625+\left(4-a\right)^{2}=1.5a^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-a\right)^{2}.

$$625+16-8a+a^{2}=1.5a^{2}$$

Add 625 and 16 to get 641.

$$641-8a+a^{2}=1.5a^{2}$$

Subtract 1.5a^{2} from both sides.

$$641-8a+a^{2}-1.5a^{2}=0$$

Combine a^{2} and -1.5a^{2} to get -0.5a^{2}.

$$641-8a-0.5a^{2}=0$$

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.

$$-0.5a^{2}-8a+641=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.5 for a, -8 for b, and 641 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-0.5\right)\times 641}}{2\left(-0.5\right)}$$

Square -8.

$$a=\frac{-\left(-8\right)±\sqrt{64-4\left(-0.5\right)\times 641}}{2\left(-0.5\right)}$$

Multiply -4 times -0.5.

$$a=\frac{-\left(-8\right)±\sqrt{64+2\times 641}}{2\left(-0.5\right)}$$

Multiply 2 times 641.

$$a=\frac{-\left(-8\right)±\sqrt{64+1282}}{2\left(-0.5\right)}$$

Add 64 to 1282.

$$a=\frac{-\left(-8\right)±\sqrt{1346}}{2\left(-0.5\right)}$$

The opposite of -8 is 8.

$$a=\frac{8±\sqrt{1346}}{2\left(-0.5\right)}$$

Multiply 2 times -0.5.

$$a=\frac{8±\sqrt{1346}}{-1}$$

Now solve the equation a=\frac{8±\sqrt{1346}}{-1} when ± is plus. Add 8 to \sqrt{1346}.

$$a=\frac{\sqrt{1346}+8}{-1}$$

Divide 8+\sqrt{1346} by -1.

$$a=-\left(\sqrt{1346}+8\right)$$

Now solve the equation a=\frac{8±\sqrt{1346}}{-1} when ± is minus. Subtract \sqrt{1346} from 8.

$$a=\frac{8-\sqrt{1346}}{-1}$$

Divide 8-\sqrt{1346} by -1.

$$a=\sqrt{1346}-8$$

The equation is now solved.

$$a=-\left(\sqrt{1346}+8\right) a=\sqrt{1346}-8$$

Steps for Completing the Square

Calculate 25 to the power of 2 and get 625.

$$625+\left(4-a\right)^{2}=1.5a^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-a\right)^{2}.

$$625+16-8a+a^{2}=1.5a^{2}$$

Add 625 and 16 to get 641.

$$641-8a+a^{2}=1.5a^{2}$$

Subtract 1.5a^{2} from both sides.

$$641-8a+a^{2}-1.5a^{2}=0$$

Combine a^{2} and -1.5a^{2} to get -0.5a^{2}.

$$641-8a-0.5a^{2}=0$$

Subtract 641 from both sides. Anything subtracted from zero gives its negation.

$$-8a-0.5a^{2}=-641$$

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.

$$-0.5a^{2}-8a=-641$$

Multiply both sides by -2.

$$\frac{-0.5a^{2}-8a}{-0.5}=-\frac{641}{-0.5}$$

Dividing by -0.5 undoes the multiplication by -0.5.

$$a^{2}+\left(-\frac{8}{-0.5}\right)a=-\frac{641}{-0.5}$$

Divide -8 by -0.5 by multiplying -8 by the reciprocal of -0.5.

$$a^{2}+16a=-\frac{641}{-0.5}$$

Divide -641 by -0.5 by multiplying -641 by the reciprocal of -0.5.

$$a^{2}+16a=1282$$

Divide 16, the coefficient of the x term, by 2 to get 8. Then add the square of 8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+16a+8^{2}=1282+8^{2}$$

Square 8.

$$a^{2}+16a+64=1282+64$$

Add 1282 to 64.

$$a^{2}+16a+64=1346$$

Factor a^{2}+16a+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+8\right)^{2}=1346$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+8\right)^{2}}=\sqrt{1346}$$

Simplify.

$$a+8=\sqrt{1346} a+8=-\sqrt{1346}$$

Subtract 8 from both sides of the equation.

$$a=\sqrt{1346}-8 a=-\sqrt{1346}-8$$

Solve for a

$$a=\sqrt{1346}-8\approx 28.687872656$$

$$a=-\sqrt{1346}-8\approx -44.687872656$$

Steps Using the Quadratic Formula

Calculate 25 to the power of 2 and get 625.

$$625+\left(4-a\right)^{2}=1.5a^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-a\right)^{2}.

$$625+16-8a+a^{2}=1.5a^{2}$$

Add 625 and 16 to get 641.

$$641-8a+a^{2}=1.5a^{2}$$

Subtract 1.5a^{2} from both sides.

$$641-8a+a^{2}-1.5a^{2}=0$$

Combine a^{2} and -1.5a^{2} to get -0.5a^{2}.

$$641-8a-0.5a^{2}=0$$

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.

$$-0.5a^{2}-8a+641=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.5 for a, -8 for b, and 641 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-0.5\right)\times 641}}{2\left(-0.5\right)}$$

Square -8.

$$a=\frac{-\left(-8\right)±\sqrt{64-4\left(-0.5\right)\times 641}}{2\left(-0.5\right)}$$

Multiply -4 times -0.5.

$$a=\frac{-\left(-8\right)±\sqrt{64+2\times 641}}{2\left(-0.5\right)}$$

Multiply 2 times 641.

$$a=\frac{-\left(-8\right)±\sqrt{64+1282}}{2\left(-0.5\right)}$$

Add 64 to 1282.

$$a=\frac{-\left(-8\right)±\sqrt{1346}}{2\left(-0.5\right)}$$

The opposite of -8 is 8.

$$a=\frac{8±\sqrt{1346}}{2\left(-0.5\right)}$$

Multiply 2 times -0.5.

$$a=\frac{8±\sqrt{1346}}{-1}$$

Now solve the equation a=\frac{8±\sqrt{1346}}{-1} when ± is plus. Add 8 to \sqrt{1346}.

$$a=\frac{\sqrt{1346}+8}{-1}$$

Divide 8+\sqrt{1346} by -1.

$$a=-\left(\sqrt{1346}+8\right)$$

Now solve the equation a=\frac{8±\sqrt{1346}}{-1} when ± is minus. Subtract \sqrt{1346} from 8.

$$a=\frac{8-\sqrt{1346}}{-1}$$

Divide 8-\sqrt{1346} by -1.

$$a=\sqrt{1346}-8$$

The equation is now solved.

$$a=-\left(\sqrt{1346}+8\right) a=\sqrt{1346}-8$$

Steps for Completing the Square

Calculate 25 to the power of 2 and get 625.

$$625+\left(4-a\right)^{2}=1.5a^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-a\right)^{2}.

$$625+16-8a+a^{2}=1.5a^{2}$$

Add 625 and 16 to get 641.

$$641-8a+a^{2}=1.5a^{2}$$

Subtract 1.5a^{2} from both sides.

$$641-8a+a^{2}-1.5a^{2}=0$$

Combine a^{2} and -1.5a^{2} to get -0.5a^{2}.

$$641-8a-0.5a^{2}=0$$

Subtract 641 from both sides. Anything subtracted from zero gives its negation.

$$-8a-0.5a^{2}=-641$$

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.

$$-0.5a^{2}-8a=-641$$

Multiply both sides by -2.

$$\frac{-0.5a^{2}-8a}{-0.5}=-\frac{641}{-0.5}$$

Dividing by -0.5 undoes the multiplication by -0.5.

$$a^{2}+\left(-\frac{8}{-0.5}\right)a=-\frac{641}{-0.5}$$

Divide -8 by -0.5 by multiplying -8 by the reciprocal of -0.5.

$$a^{2}+16a=-\frac{641}{-0.5}$$

Divide -641 by -0.5 by multiplying -641 by the reciprocal of -0.5.

$$a^{2}+16a=1282$$

Divide 16, the coefficient of the x term, by 2 to get 8. Then add the square of 8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+16a+8^{2}=1282+8^{2}$$

Square 8.

$$a^{2}+16a+64=1282+64$$

Add 1282 to 64.

$$a^{2}+16a+64=1346$$

Factor a^{2}+16a+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+8\right)^{2}=1346$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+8\right)^{2}}=\sqrt{1346}$$

Simplify.

$$a+8=\sqrt{1346} a+8=-\sqrt{1346}$$

Subtract 8 from both sides of the equation.

$$a=\sqrt{1346}-8 a=-\sqrt{1346}-8$$