Solve 4w^2=-13w-3

$$4 w ^ { 2 } = - 13 w - 3$$

Solve for w

$$w=-3$$

$$w=-\frac{1}{4}=-0.25$$

Steps Using Factoring By Grouping

Add 13w to both sides.

$$4w^{2}+13w=-3$$

Add 3 to both sides.

$$4w^{2}+13w+3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4w^{2}+aw+bw+3. To find a and b, set up a system to be solved.

$$a+b=13 ab=4\times 3=12$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 13.

$$a=1 b=12$$

Rewrite 4w^{2}+13w+3 as \left(4w^{2}+w\right)+\left(12w+3\right).

$$\left(4w^{2}+w\right)+\left(12w+3\right)$$

Factor out w in the first and 3 in the second group.

$$w\left(4w+1\right)+3\left(4w+1\right)$$

Factor out common term 4w+1 by using distributive property.

$$\left(4w+1\right)\left(w+3\right)$$

To find equation solutions, solve 4w+1=0 and w+3=0.

$$w=-\frac{1}{4} w=-3$$

Steps Using the Quadratic Formula

Add 13w to both sides.

$$4w^{2}+13w=-3$$

Add 3 to both sides.

$$4w^{2}+13w+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 13 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-13±\sqrt{13^{2}-4\times 4\times 3}}{2\times 4}$$

Square 13.

$$w=\frac{-13±\sqrt{169-4\times 4\times 3}}{2\times 4}$$

Multiply -4 times 4.

$$w=\frac{-13±\sqrt{169-16\times 3}}{2\times 4}$$

Multiply -16 times 3.

$$w=\frac{-13±\sqrt{169-48}}{2\times 4}$$

Add 169 to -48.

$$w=\frac{-13±\sqrt{121}}{2\times 4}$$

Take the square root of 121.

$$w=\frac{-13±11}{2\times 4}$$

Multiply 2 times 4.

$$w=\frac{-13±11}{8}$$

Now solve the equation w=\frac{-13±11}{8} when ± is plus. Add -13 to 11.

$$w=-\frac{2}{8}$$

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

$$w=-\frac{1}{4}$$

Now solve the equation w=\frac{-13±11}{8} when ± is minus. Subtract 11 from -13.

$$w=-\frac{24}{8}$$

Divide -24 by 8.

$$w=-3$$

The equation is now solved.

$$w=-\frac{1}{4} w=-3$$

Steps for Completing the Square

Add 13w to both sides.

$$4w^{2}+13w=-3$$

Divide both sides by 4.

$$\frac{4w^{2}+13w}{4}=-\frac{3}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$w^{2}+\frac{13}{4}w=-\frac{3}{4}$$

Divide \frac{13}{4}, the coefficient of the x term, by 2 to get \frac{13}{8}. Then add the square of \frac{13}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}+\frac{13}{4}w+\left(\frac{13}{8}\right)^{2}=-\frac{3}{4}+\left(\frac{13}{8}\right)^{2}$$

Square \frac{13}{8} by squaring both the numerator and the denominator of the fraction.

$$w^{2}+\frac{13}{4}w+\frac{169}{64}=-\frac{3}{4}+\frac{169}{64}$$

Add -\frac{3}{4} to \frac{169}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$w^{2}+\frac{13}{4}w+\frac{169}{64}=\frac{121}{64}$$

Factor w^{2}+\frac{13}{4}w+\frac{169}{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}.

$$\left(w+\frac{13}{8}\right)^{2}=\frac{121}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w+\frac{13}{8}\right)^{2}}=\sqrt{\frac{121}{64}}$$

Simplify.

$$w+\frac{13}{8}=\frac{11}{8} w+\frac{13}{8}=-\frac{11}{8}$$

Subtract \frac{13}{8} from both sides of the equation.

$$w=-\frac{1}{4} w=-3$$