Solve P(x)=-21x^2-8x+5

$$P ( x ) = - 21 x ^ { 2 } - 8 x + 5$$

Factor

$$-\left(3x-1\right)\left(7x+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -21x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=-8 ab=-21\times 5=-105$$

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 -105.

$$1,-105 3,-35 5,-21 7,-15$$

Calculate the sum for each pair.

$$1-105=-104 3-35=-32 5-21=-16 7-15=-8$$

The solution is the pair that gives sum -8.

$$a=7 b=-15$$

Rewrite -21x^{2}-8x+5 as \left(-21x^{2}+7x\right)+\left(-15x+5\right).

$$\left(-21x^{2}+7x\right)+\left(-15x+5\right)$$

Factor out -7x in the first and -5 in the second group.

$$-7x\left(3x-1\right)-5\left(3x-1\right)$$

Factor out common term 3x-1 by using distributive property.

$$\left(3x-1\right)\left(-7x-5\right)$$

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.

$$-21x^{2}-8x+5=0$$

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.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-21\right)\times 5}}{2\left(-21\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-21\right)\times 5}}{2\left(-21\right)}$$

Multiply -4 times -21.

$$x=\frac{-\left(-8\right)±\sqrt{64+84\times 5}}{2\left(-21\right)}$$

Multiply 84 times 5.

$$x=\frac{-\left(-8\right)±\sqrt{64+420}}{2\left(-21\right)}$$

Add 64 to 420.

$$x=\frac{-\left(-8\right)±\sqrt{484}}{2\left(-21\right)}$$

Take the square root of 484.

$$x=\frac{-\left(-8\right)±22}{2\left(-21\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±22}{2\left(-21\right)}$$

Multiply 2 times -21.

$$x=\frac{8±22}{-42}$$

Now solve the equation x=\frac{8±22}{-42} when ± is plus. Add 8 to 22.

$$x=\frac{30}{-42}$$

Reduce the fraction \frac{30}{-42} to lowest terms by extracting and canceling out 6.

$$x=-\frac{5}{7}$$

Now solve the equation x=\frac{8±22}{-42} when ± is minus. Subtract 22 from 8.

$$x=-\frac{14}{-42}$$

Reduce the fraction \frac{-14}{-42} to lowest terms by extracting and canceling out 14.

$$x=\frac{1}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{5}{7} for x_{1} and \frac{1}{3} for x_{2}.

$$-21x^{2}-8x+5=-21\left(x-\left(-\frac{5}{7}\right)\right)\left(x-\frac{1}{3}\right)$$

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

$$-21x^{2}-8x+5=-21\left(x+\frac{5}{7}\right)\left(x-\frac{1}{3}\right)$$

Add \frac{5}{7} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-21x^{2}-8x+5=-21\times \frac{-7x-5}{-7}\left(x-\frac{1}{3}\right)$$

Subtract \frac{1}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-21x^{2}-8x+5=-21\times \frac{-7x-5}{-7}\times \frac{-3x+1}{-3}$$

Multiply \frac{-7x-5}{-7} times \frac{-3x+1}{-3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-21x^{2}-8x+5=-21\times \frac{\left(-7x-5\right)\left(-3x+1\right)}{-7\left(-3\right)}$$

Multiply -7 times -3.

$$-21x^{2}-8x+5=-21\times \frac{\left(-7x-5\right)\left(-3x+1\right)}{21}$$

Cancel out 21, the greatest common factor in -21 and 21.

$$-21x^{2}-8x+5=-\left(-7x-5\right)\left(-3x+1\right)$$

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.

$$x ^ 2 +\frac{8}{21}x -\frac{5}{21} = 0$$

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

$$r + s = -\frac{8}{21} rs = -\frac{5}{21}$$

Two numbers r and s sum up to -\frac{8}{21} exactly when the average of the two numbers is \frac{1}{2}*-\frac{8}{21} = -\frac{4}{21}. 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>

$$r = -\frac{4}{21} - u s = -\frac{4}{21} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{5}{21}

$$(-\frac{4}{21} - u) (-\frac{4}{21} + u) = -\frac{5}{21}$$

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

$$\frac{16}{441} - u^2 = -\frac{5}{21}$$

Simplify the expression by subtracting \frac{16}{441} on both sides

$$-u^2 = -\frac{5}{21}-\frac{16}{441} = -\frac{121}{441}$$

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

$$u^2 = \frac{121}{441} u = \pm\sqrt{\frac{121}{441}} = \pm \frac{11}{21}$$

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

$$r =-\frac{4}{21} - \frac{11}{21} = -0.714 s = -\frac{4}{21} + \frac{11}{21} = 0.333$$

Evaluate

$$5-8x-21x^{2}$$