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

Subtract 49x^{2} from 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.

$$a+b=28 ab=-49\left(-4\right)=196$$

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.

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

To find equation solutions, solve 7x-2=0 and -7x+2=0.

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

Steps Using the Quadratic Formula

Subtract 49x^{2} from 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}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-49\right)\left(-4\right)}}{2\left(-49\right)}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-49\right)\left(-4\right)}}{2\left(-49\right)}$$

Multiply -4 times -49.

$$x=\frac{-28±\sqrt{784+196\left(-4\right)}}{2\left(-49\right)}$$

Multiply 196 times -4.

$$x=\frac{-28±\sqrt{784-784}}{2\left(-49\right)}$$

Add 784 to -784.

$$x=\frac{-28±\sqrt{0}}{2\left(-49\right)}$$

Take the square root of 0.

$$x=-\frac{28}{2\left(-49\right)}$$

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

Subtract 49x^{2} from both sides.

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

Add 4 to both sides. Anything plus zero gives itself.

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

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 4 by -49.

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

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.

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

Add \frac{2}{7} to both sides of the equation.

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

The equation is now solved. Solutions are the same.

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