Solve 28x+4=-49x^2

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

Solve for x

$$x=-\frac{2}{7}\approx -0.285714286$$

Steps Using Factoring By Grouping

Add 49x^{2} to both sides.

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

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

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

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

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

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

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

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

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

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

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

Rewrite as a binomial square.

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

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

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

Steps Using the Quadratic Formula

Add 49x^{2} to both sides.

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

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.

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

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

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

Square 28.

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

Multiply -4 times 49.

$$x=\frac{-28±\sqrt{784-196\times 4}}{2\times 49}$$

Multiply -196 times 4.

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

Add 784 to -784.

$$x=\frac{-28±\sqrt{0}}{2\times 49}$$

Take the square root of 0.

$$x=-\frac{28}{2\times 49}$$

Multiply 2 times 49.

$$x=-\frac{28}{98}$$

Reduce the fraction \frac{-28}{98} to lowest terms by extracting and canceling out 14.

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

Steps for Completing the Square

Add 49x^{2} to both sides.

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

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

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

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.

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

Divide both sides by 49.

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

Dividing by 49 undoes the multiplication by 49.

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

Reduce the fraction \frac{28}{49} to lowest terms by extracting and canceling out 7.

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

Divide \frac{4}{7}, the coefficient of the x term, by 2 to get \frac{2}{7}. Then add the square of \frac{2}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

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

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

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

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

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

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

Take the square root of both sides of the equation.

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

Simplify.

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

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

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

The equation is now solved. Solutions are the same.

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