Solve 8b^2-87b+70

$$8 b ^ { 2 } - 87 b + 70$$

Factor

$$\left(b-10\right)\left(8b-7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 8b^{2}+pb+qb+70. To find p and q, set up a system to be solved.

$$p+q=-87 pq=8\times 70=560$$

Since pq is positive, p and q have the same sign. Since p+q is negative, p and q are both negative. List all such integer pairs that give product 560.

$$-1,-560 -2,-280 -4,-140 -5,-112 -7,-80 -8,-70 -10,-56 -14,-40 -16,-35 -20,-28$$

Calculate the sum for each pair.

$$-1-560=-561 -2-280=-282 -4-140=-144 -5-112=-117 -7-80=-87 -8-70=-78 -10-56=-66 -14-40=-54 -16-35=-51 -20-28=-48$$

The solution is the pair that gives sum -87.

$$p=-80 q=-7$$

Rewrite 8b^{2}-87b+70 as \left(8b^{2}-80b\right)+\left(-7b+70\right).

$$\left(8b^{2}-80b\right)+\left(-7b+70\right)$$

Factor out 8b in the first and -7 in the second group.

$$8b\left(b-10\right)-7\left(b-10\right)$$

Factor out common term b-10 by using distributive property.

$$\left(b-10\right)\left(8b-7\right)$$

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.

$$8b^{2}-87b+70=0$$

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.

$$b=\frac{-\left(-87\right)±\sqrt{\left(-87\right)^{2}-4\times 8\times 70}}{2\times 8}$$

Square -87.

$$b=\frac{-\left(-87\right)±\sqrt{7569-4\times 8\times 70}}{2\times 8}$$

Multiply -4 times 8.

$$b=\frac{-\left(-87\right)±\sqrt{7569-32\times 70}}{2\times 8}$$

Multiply -32 times 70.

$$b=\frac{-\left(-87\right)±\sqrt{7569-2240}}{2\times 8}$$

Add 7569 to -2240.

$$b=\frac{-\left(-87\right)±\sqrt{5329}}{2\times 8}$$

Take the square root of 5329.

$$b=\frac{-\left(-87\right)±73}{2\times 8}$$

The opposite of -87 is 87.

$$b=\frac{87±73}{2\times 8}$$

Multiply 2 times 8.

$$b=\frac{87±73}{16}$$

Now solve the equation b=\frac{87±73}{16} when ± is plus. Add 87 to 73.

$$b=\frac{160}{16}$$

Divide 160 by 16.

$$b=10$$

Now solve the equation b=\frac{87±73}{16} when ± is minus. Subtract 73 from 87.

$$b=\frac{14}{16}$$

Reduce the fraction \frac{14}{16} to lowest terms by extracting and canceling out 2.

$$b=\frac{7}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 10 for x_{1} and \frac{7}{8} for x_{2}.

$$8b^{2}-87b+70=8\left(b-10\right)\left(b-\frac{7}{8}\right)$$

Subtract \frac{7}{8} from b by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$8b^{2}-87b+70=8\left(b-10\right)\times \frac{8b-7}{8}$$

Cancel out 8, the greatest common factor in 8 and 8.

$$8b^{2}-87b+70=\left(b-10\right)\left(8b-7\right)$$

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.This is achieved by dividing both sides of the equation by 8

$$x ^ 2 -\frac{87}{8}x +\frac{35}{4} = 0$$

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

$$r + s = \frac{87}{8} rs = \frac{35}{4}$$

Two numbers r and s sum up to \frac{87}{8} exactly when the average of the two numbers is \frac{1}{2}*\frac{87}{8} = \frac{87}{16}. 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>

$$r = \frac{87}{16} - u s = \frac{87}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{35}{4}

$$(\frac{87}{16} - u) (\frac{87}{16} + u) = \frac{35}{4}$$

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

$$\frac{7569}{256} - u^2 = \frac{35}{4}$$

Simplify the expression by subtracting \frac{7569}{256} on both sides

$$-u^2 = \frac{35}{4}-\frac{7569}{256} = -\frac{5329}{256}$$

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

$$u^2 = \frac{5329}{256} u = \pm\sqrt{\frac{5329}{256}} = \pm \frac{73}{16}$$

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

$$r =\frac{87}{16} - \frac{73}{16} = 0.875 s = \frac{87}{16} + \frac{73}{16} = 10$$

Evaluate

$$\left(b-10\right)\left(8b-7\right)$$