Solve -a^2-2a+3/1-a=3/4

$$\frac { - a ^ { 2 } - 2 a + 3 } { 1 - a } = \frac { 3 } { 4 }$$

Solve for a

$$a = -\frac{9}{4} = -2\frac{1}{4} = -2.25$$

Steps Using Factoring By Grouping

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 4\left(a-1\right), the least common multiple of 1-a,4.

$$-4\left(-a^{2}-2a+3\right)=3\left(a-1\right)$$

Use the distributive property to multiply -4 by -a^{2}-2a+3.

$$-4\left(-a^{2}\right)+8a-12=3\left(a-1\right)$$

Multiply -4 and -1 to get 4.

$$4a^{2}+8a-12=3\left(a-1\right)$$

Use the distributive property to multiply 3 by a-1.

$$4a^{2}+8a-12=3a-3$$

Subtract 3a from both sides.

$$4a^{2}+8a-12-3a=-3$$

Combine 8a and -3a to get 5a.

$$4a^{2}+5a-12=-3$$

Add 3 to both sides.

$$4a^{2}+5a-12+3=0$$

Add -12 and 3 to get -9.

$$4a^{2}+5a-9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4a^{2}+aa+ba-9. To find a and b, set up a system to be solved.

$$a+b=5 ab=4\left(-9\right)=-36$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -36.

$$-1,36 -2,18 -3,12 -4,9 -6,6$$

Calculate the sum for each pair.

$$-1+36=35 -2+18=16 -3+12=9 -4+9=5 -6+6=0$$

The solution is the pair that gives sum 5.

$$a=-4 b=9$$

Rewrite 4a^{2}+5a-9 as \left(4a^{2}-4a\right)+\left(9a-9\right).

$$\left(4a^{2}-4a\right)+\left(9a-9\right)$$

Factor out 4a in the first and 9 in the second group.

$$4a\left(a-1\right)+9\left(a-1\right)$$

Factor out common term a-1 by using distributive property.

$$\left(a-1\right)\left(4a+9\right)$$

To find equation solutions, solve a-1=0 and 4a+9=0.

$$a=1 a=-\frac{9}{4}$$

Variable a cannot be equal to 1.

$$a=-\frac{9}{4}$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 4\left(a-1\right), the least common multiple of 1-a,4.

$$-4\left(-a^{2}-2a+3\right)=3\left(a-1\right)$$

Use the distributive property to multiply -4 by -a^{2}-2a+3.

$$-4\left(-a^{2}\right)+8a-12=3\left(a-1\right)$$

Multiply -4 and -1 to get 4.

$$4a^{2}+8a-12=3\left(a-1\right)$$

Use the distributive property to multiply 3 by a-1.

$$4a^{2}+8a-12=3a-3$$

Subtract 3a from both sides.

$$4a^{2}+8a-12-3a=-3$$

Combine 8a and -3a to get 5a.

$$4a^{2}+5a-12=-3$$

Add 3 to both sides.

$$4a^{2}+5a-12+3=0$$

Add -12 and 3 to get -9.

$$4a^{2}+5a-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 5 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-5±\sqrt{5^{2}-4\times 4\left(-9\right)}}{2\times 4}$$

Square 5.

$$a=\frac{-5±\sqrt{25-4\times 4\left(-9\right)}}{2\times 4}$$

Multiply -4 times 4.

$$a=\frac{-5±\sqrt{25-16\left(-9\right)}}{2\times 4}$$

Multiply -16 times -9.

$$a=\frac{-5±\sqrt{25+144}}{2\times 4}$$

Add 25 to 144.

$$a=\frac{-5±\sqrt{169}}{2\times 4}$$

Take the square root of 169.

$$a=\frac{-5±13}{2\times 4}$$

Multiply 2 times 4.

$$a=\frac{-5±13}{8}$$

Now solve the equation a=\frac{-5±13}{8} when ± is plus. Add -5 to 13.

$$a=\frac{8}{8}$$

Divide 8 by 8.

$$a=1$$

Now solve the equation a=\frac{-5±13}{8} when ± is minus. Subtract 13 from -5.

$$a=-\frac{18}{8}$$

Reduce the fraction \frac{-18}{8} to lowest terms by extracting and canceling out 2.

$$a=-\frac{9}{4}$$

The equation is now solved.

$$a=1 a=-\frac{9}{4}$$

Variable a cannot be equal to 1.

$$a=-\frac{9}{4}$$

Steps for Completing the Square

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 4\left(a-1\right), the least common multiple of 1-a,4.

$$-4\left(-a^{2}-2a+3\right)=3\left(a-1\right)$$

Use the distributive property to multiply -4 by -a^{2}-2a+3.

$$-4\left(-a^{2}\right)+8a-12=3\left(a-1\right)$$

Multiply -4 and -1 to get 4.

$$4a^{2}+8a-12=3\left(a-1\right)$$

Use the distributive property to multiply 3 by a-1.

$$4a^{2}+8a-12=3a-3$$

Subtract 3a from both sides.

$$4a^{2}+8a-12-3a=-3$$

Combine 8a and -3a to get 5a.

$$4a^{2}+5a-12=-3$$

Add 12 to both sides.

$$4a^{2}+5a=-3+12$$

Add -3 and 12 to get 9.

$$4a^{2}+5a=9$$

Divide both sides by 4.

$$\frac{4a^{2}+5a}{4}=\frac{9}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$a^{2}+\frac{5}{4}a=\frac{9}{4}$$

Divide \frac{5}{4}, the coefficient of the x term, by 2 to get \frac{5}{8}. Then add the square of \frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{5}{4}a+\left(\frac{5}{8}\right)^{2}=\frac{9}{4}+\left(\frac{5}{8}\right)^{2}$$

Square \frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{5}{4}a+\frac{25}{64}=\frac{9}{4}+\frac{25}{64}$$

Add \frac{9}{4} to \frac{25}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}+\frac{5}{4}a+\frac{25}{64}=\frac{169}{64}$$

Factor a^{2}+\frac{5}{4}a+\frac{25}{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}.

$$\left(a+\frac{5}{8}\right)^{2}=\frac{169}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{5}{8}\right)^{2}}=\sqrt{\frac{169}{64}}$$

Simplify.

$$a+\frac{5}{8}=\frac{13}{8} a+\frac{5}{8}=-\frac{13}{8}$$

Subtract \frac{5}{8} from both sides of the equation.

$$a=1 a=-\frac{9}{4}$$

Variable a cannot be equal to 1.

$$a=-\frac{9}{4}$$