Solve 56-2a/2=56-a/56-2a

$$\frac { 56 - 2 a } { 2 } = \frac { 56 - a } { 56 - 2 a }$$

Solve for a

$$a=24$$

$$a = \frac{63}{2} = 31\frac{1}{2} = 31.5$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 28 since division by zero is not defined. Multiply both sides of the equation by 2\left(a-28\right), the least common multiple of 2,56-2a.

$$\left(a-28\right)\left(56-2a\right)=-\left(56-a\right)$$

Use the distributive property to multiply a-28 by 56-2a and combine like terms.

$$112a-2a^{2}-1568=-\left(56-a\right)$$

To find the opposite of 56-a, find the opposite of each term.

$$112a-2a^{2}-1568=-56+a$$

Subtract -56 from both sides.

$$112a-2a^{2}-1568-\left(-56\right)=a$$

The opposite of -56 is 56.

$$112a-2a^{2}-1568+56=a$$

Subtract a from both sides.

$$112a-2a^{2}-1568+56-a=0$$

Add -1568 and 56 to get -1512.

$$112a-2a^{2}-1512-a=0$$

Combine 112a and -a to get 111a.

$$111a-2a^{2}-1512=0$$

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.

$$-2a^{2}+111a-1512=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 111 for b, and -1512 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-111±\sqrt{111^{2}-4\left(-2\right)\left(-1512\right)}}{2\left(-2\right)}$$

Square 111.

$$a=\frac{-111±\sqrt{12321-4\left(-2\right)\left(-1512\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$a=\frac{-111±\sqrt{12321+8\left(-1512\right)}}{2\left(-2\right)}$$

Multiply 8 times -1512.

$$a=\frac{-111±\sqrt{12321-12096}}{2\left(-2\right)}$$

Add 12321 to -12096.

$$a=\frac{-111±\sqrt{225}}{2\left(-2\right)}$$

Take the square root of 225.

$$a=\frac{-111±15}{2\left(-2\right)}$$

Multiply 2 times -2.

$$a=\frac{-111±15}{-4}$$

Now solve the equation a=\frac{-111±15}{-4} when ± is plus. Add -111 to 15.

$$a=-\frac{96}{-4}$$

Divide -96 by -4.

$$a=24$$

Now solve the equation a=\frac{-111±15}{-4} when ± is minus. Subtract 15 from -111.

$$a=-\frac{126}{-4}$$

Reduce the fraction \frac{-126}{-4} to lowest terms by extracting and canceling out 2.

$$a=\frac{63}{2}$$

The equation is now solved.

$$a=24 a=\frac{63}{2}$$

Steps for Completing the Square

Variable a cannot be equal to 28 since division by zero is not defined. Multiply both sides of the equation by 2\left(a-28\right), the least common multiple of 2,56-2a.

$$\left(a-28\right)\left(56-2a\right)=-\left(56-a\right)$$

Use the distributive property to multiply a-28 by 56-2a and combine like terms.

$$112a-2a^{2}-1568=-\left(56-a\right)$$

To find the opposite of 56-a, find the opposite of each term.

$$112a-2a^{2}-1568=-56+a$$

Subtract a from both sides.

$$112a-2a^{2}-1568-a=-56$$

Combine 112a and -a to get 111a.

$$111a-2a^{2}-1568=-56$$

Add 1568 to both sides.

$$111a-2a^{2}=-56+1568$$

Add -56 and 1568 to get 1512.

$$111a-2a^{2}=1512$$

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.

$$-2a^{2}+111a=1512$$

Divide both sides by -2.

$$\frac{-2a^{2}+111a}{-2}=\frac{1512}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$a^{2}+\frac{111}{-2}a=\frac{1512}{-2}$$

Divide 111 by -2.

$$a^{2}-\frac{111}{2}a=\frac{1512}{-2}$$

Divide 1512 by -2.

$$a^{2}-\frac{111}{2}a=-756$$

Divide -\frac{111}{2}, the coefficient of the x term, by 2 to get -\frac{111}{4}. Then add the square of -\frac{111}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{111}{2}a+\left(-\frac{111}{4}\right)^{2}=-756+\left(-\frac{111}{4}\right)^{2}$$

Square -\frac{111}{4} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{111}{2}a+\frac{12321}{16}=-756+\frac{12321}{16}$$

Add -756 to \frac{12321}{16}.

$$a^{2}-\frac{111}{2}a+\frac{12321}{16}=\frac{225}{16}$$

Factor a^{2}-\frac{111}{2}a+\frac{12321}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{111}{4}\right)^{2}=\frac{225}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{111}{4}\right)^{2}}=\sqrt{\frac{225}{16}}$$

Simplify.

$$a-\frac{111}{4}=\frac{15}{4} a-\frac{111}{4}=-\frac{15}{4}$$

Add \frac{111}{4} to both sides of the equation.

$$a=\frac{63}{2} a=24$$