Solve -49x^2-4+28x

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

Factor

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

Solution Steps

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

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

Factor the expression by grouping. First, the expression 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.

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

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

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

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.

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

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.

$$x=\frac{-28±0}{2\left(-49\right)}$$

Multiply 2 times -49.

$$x=\frac{-28±0}{-98}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2}{7} for x_{1} and \frac{2}{7} for x_{2}.

$$-49x^{2}+28x-4=-49\left(x-\frac{2}{7}\right)\left(x-\frac{2}{7}\right)$$

Subtract \frac{2}{7} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-49x^{2}+28x-4=-49\times \frac{-7x+2}{-7}\left(x-\frac{2}{7}\right)$$

Subtract \frac{2}{7} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-49x^{2}+28x-4=-49\times \frac{-7x+2}{-7}\times \frac{-7x+2}{-7}$$

Multiply \frac{-7x+2}{-7} times \frac{-7x+2}{-7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-49x^{2}+28x-4=-49\times \frac{\left(-7x+2\right)\left(-7x+2\right)}{-7\left(-7\right)}$$

Multiply -7 times -7.

$$-49x^{2}+28x-4=-49\times \frac{\left(-7x+2\right)\left(-7x+2\right)}{49}$$

Cancel out 49, the greatest common factor in -49 and 49.

$$-49x^{2}+28x-4=-\left(-7x+2\right)\left(-7x+2\right)$$

Evaluate

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