Solve -4x^2-7x+2

$$- 4 x ^ { 2 } - 7 x + 2$$

Factor

$$-\left(4x-1\right)\left(x+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -4x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=-7 ab=-4\times 2=-8$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -8.

$$1,-8 2,-4$$

Calculate the sum for each pair.

$$1-8=-7 2-4=-2$$

The solution is the pair that gives sum -7.

$$a=1 b=-8$$

Rewrite -4x^{2}-7x+2 as \left(-4x^{2}+x\right)+\left(-8x+2\right).

$$\left(-4x^{2}+x\right)+\left(-8x+2\right)$$

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

$$-x\left(4x-1\right)-2\left(4x-1\right)$$

Factor out common term 4x-1 by using distributive property.

$$\left(4x-1\right)\left(-x-2\right)$$

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.

$$-4x^{2}-7x+2=0$$

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.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-4\right)\times 2}}{2\left(-4\right)}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-4\right)\times 2}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-7\right)±\sqrt{49+16\times 2}}{2\left(-4\right)}$$

Multiply 16 times 2.

$$x=\frac{-\left(-7\right)±\sqrt{49+32}}{2\left(-4\right)}$$

Add 49 to 32.

$$x=\frac{-\left(-7\right)±\sqrt{81}}{2\left(-4\right)}$$

Take the square root of 81.

$$x=\frac{-\left(-7\right)±9}{2\left(-4\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±9}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{7±9}{-8}$$

Now solve the equation x=\frac{7±9}{-8} when ± is plus. Add 7 to 9.

$$x=\frac{16}{-8}$$

Divide 16 by -8.

$$x=-2$$

Now solve the equation x=\frac{7±9}{-8} when ± is minus. Subtract 9 from 7.

$$x=-\frac{2}{-8}$$

Reduce the fraction \frac{-2}{-8} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -2 for x_{1} and \frac{1}{4} for x_{2}.

$$-4x^{2}-7x+2=-4\left(x-\left(-2\right)\right)\left(x-\frac{1}{4}\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-4x^{2}-7x+2=-4\left(x+2\right)\left(x-\frac{1}{4}\right)$$

Subtract \frac{1}{4} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-4x^{2}-7x+2=-4\left(x+2\right)\times \frac{-4x+1}{-4}$$

Cancel out 4, the greatest common factor in -4 and 4.

$$-4x^{2}-7x+2=\left(x+2\right)\left(-4x+1\right)$$

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.

$$x ^ 2 +\frac{7}{4}x -\frac{1}{2} = 0$$

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

$$r + s = -\frac{7}{4} rs = -\frac{1}{2}$$

Two numbers r and s sum up to -\frac{7}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{7}{4} = -\frac{7}{8}. 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>

$$r = -\frac{7}{8} - u s = -\frac{7}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{2}

$$(-\frac{7}{8} - u) (-\frac{7}{8} + u) = -\frac{1}{2}$$

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

$$\frac{49}{64} - u^2 = -\frac{1}{2}$$

Simplify the expression by subtracting \frac{49}{64} on both sides

$$-u^2 = -\frac{1}{2}-\frac{49}{64} = -\frac{81}{64}$$

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

$$u^2 = \frac{81}{64} u = \pm\sqrt{\frac{81}{64}} = \pm \frac{9}{8}$$

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

$$r =-\frac{7}{8} - \frac{9}{8} = -2 s = -\frac{7}{8} + \frac{9}{8} = 0.250$$

Evaluate

$$-\left(4x-1\right)\left(x+2\right)$$