Solve 4w^2+49=-28w

$$4 w ^ { 2 } + 49 = - 28 w$$

Solve for w

$$w = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

Steps Using Factoring By Grouping

Add 28w to both sides.

$$4w^{2}+49+28w=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$4w^{2}+28w+49=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4w^{2}+aw+bw+49. To find a and b, set up a system to be solved.

$$a+b=28 ab=4\times 49=196$$

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 196.

$$1,196 2,98 4,49 7,28 14,14$$

Calculate the sum for each pair.

$$1+196=197 2+98=100 4+49=53 7+28=35 14+14=28$$

The solution is the pair that gives sum 28.

$$a=14 b=14$$

Rewrite 4w^{2}+28w+49 as \left(4w^{2}+14w\right)+\left(14w+49\right).

$$\left(4w^{2}+14w\right)+\left(14w+49\right)$$

Factor out 2w in the first and 7 in the second group.

$$2w\left(2w+7\right)+7\left(2w+7\right)$$

Factor out common term 2w+7 by using distributive property.

$$\left(2w+7\right)\left(2w+7\right)$$

Rewrite as a binomial square.

$$\left(2w+7\right)^{2}$$

To find equation solution, solve 2w+7=0.

$$w=-\frac{7}{2}$$

Steps Using the Quadratic Formula

Add 28w to both sides.

$$4w^{2}+49+28w=0$$

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.

$$4w^{2}+28w+49=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 28 for b, and 49 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-28±\sqrt{28^{2}-4\times 4\times 49}}{2\times 4}$$

Square 28.

$$w=\frac{-28±\sqrt{784-4\times 4\times 49}}{2\times 4}$$

Multiply -4 times 4.

$$w=\frac{-28±\sqrt{784-16\times 49}}{2\times 4}$$

Multiply -16 times 49.

$$w=\frac{-28±\sqrt{784-784}}{2\times 4}$$

Add 784 to -784.

$$w=\frac{-28±\sqrt{0}}{2\times 4}$$

Take the square root of 0.

$$w=-\frac{28}{2\times 4}$$

Multiply 2 times 4.

$$w=-\frac{28}{8}$$

Reduce the fraction \frac{-28}{8} to lowest terms by extracting and canceling out 4.

$$w=-\frac{7}{2}$$

Steps for Completing the Square

Add 28w to both sides.

$$4w^{2}+49+28w=0$$

Subtract 49 from both sides. Anything subtracted from zero gives its negation.

$$4w^{2}+28w=-49$$

Divide both sides by 4.

$$\frac{4w^{2}+28w}{4}=-\frac{49}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$w^{2}+\frac{28}{4}w=-\frac{49}{4}$$

Divide 28 by 4.

$$w^{2}+7w=-\frac{49}{4}$$

Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}+7w+\left(\frac{7}{2}\right)^{2}=-\frac{49}{4}+\left(\frac{7}{2}\right)^{2}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$w^{2}+7w+\frac{49}{4}=\frac{-49+49}{4}$$

Add -\frac{49}{4} to \frac{49}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$w^{2}+7w+\frac{49}{4}=0$$

Factor w^{2}+7w+\frac{49}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(w+\frac{7}{2}\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w+\frac{7}{2}\right)^{2}}=\sqrt{0}$$

Simplify.

$$w+\frac{7}{2}=0 w+\frac{7}{2}=0$$

Subtract \frac{7}{2} from both sides of the equation.

$$w=-\frac{7}{2} w=-\frac{7}{2}$$

The equation is now solved. Solutions are the same.

$$w=-\frac{7}{2}$$