Solve -98x^2+28x-2

$$- 98 x ^ { 2 } + 28 x - 2$$

Factor

$$-2\left(7x-1\right)^{2}$$

Solution Steps

Factor out 2.

$$2\left(-49x^{2}+14x-1\right)$$

Consider -49x^{2}+14x-1. Factor the expression by grouping. First, the expression needs to be rewritten as -49x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=14 ab=-49\left(-1\right)=49$$

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 49.

$$1,49 7,7$$

Calculate the sum for each pair.

$$1+49=50 7+7=14$$

The solution is the pair that gives sum 14.

$$a=7 b=7$$

Rewrite -49x^{2}+14x-1 as \left(-49x^{2}+7x\right)+\left(7x-1\right).

$$\left(-49x^{2}+7x\right)+\left(7x-1\right)$$

Factor out -7x in -49x^{2}+7x.

$$-7x\left(7x-1\right)+7x-1$$

Factor out common term 7x-1 by using distributive property.

$$\left(7x-1\right)\left(-7x+1\right)$$

Rewrite the complete factored expression.

$$2\left(7x-1\right)\left(-7x+1\right)$$

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.

$$-98x^{2}+28x-2=0$$

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.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-98\right)\left(-2\right)}}{2\left(-98\right)}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-98\right)\left(-2\right)}}{2\left(-98\right)}$$

Multiply -4 times -98.

$$x=\frac{-28±\sqrt{784+392\left(-2\right)}}{2\left(-98\right)}$$

Multiply 392 times -2.

$$x=\frac{-28±\sqrt{784-784}}{2\left(-98\right)}$$

Add 784 to -784.

$$x=\frac{-28±\sqrt{0}}{2\left(-98\right)}$$

Take the square root of 0.

$$x=\frac{-28±0}{2\left(-98\right)}$$

Multiply 2 times -98.

$$x=\frac{-28±0}{-196}$$

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

$$-98x^{2}+28x-2=-98\left(x-\frac{1}{7}\right)\left(x-\frac{1}{7}\right)$$

Subtract \frac{1}{7} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-98x^{2}+28x-2=-98\times \frac{-7x+1}{-7}\left(x-\frac{1}{7}\right)$$

Subtract \frac{1}{7} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-98x^{2}+28x-2=-98\times \frac{-7x+1}{-7}\times \frac{-7x+1}{-7}$$

Multiply \frac{-7x+1}{-7} times \frac{-7x+1}{-7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-98x^{2}+28x-2=-98\times \frac{\left(-7x+1\right)\left(-7x+1\right)}{-7\left(-7\right)}$$

Multiply -7 times -7.

$$-98x^{2}+28x-2=-98\times \frac{\left(-7x+1\right)\left(-7x+1\right)}{49}$$

Cancel out 49, the greatest common factor in -98 and 49.

$$-98x^{2}+28x-2=-2\left(-7x+1\right)\left(-7x+1\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 -\frac{2}{7}x +\frac{1}{49} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = \frac{2}{7} rs = \frac{1}{49}$$

Two numbers r and s sum up to \frac{2}{7} exactly when the average of the two numbers is \frac{1}{2}*\frac{2}{7} = \frac{1}{7}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{1}{7} - u s = \frac{1}{7} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{49}

$$(\frac{1}{7} - u) (\frac{1}{7} + u) = \frac{1}{49}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{1}{49} - u^2 = \frac{1}{49}$$

Simplify the expression by subtracting \frac{1}{49} on both sides

$$-u^2 = \frac{1}{49}-\frac{1}{49} = 0$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 0 u = 0$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r = s = \frac{1}{7} = 0.143$$

Evaluate

$$-2\left(7x-1\right)^{2}$$