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Section 2.6 : Quadratic Equations - Part II
10. Use the quadratic formula to solve the following quadratic equation.
\[9{w^2} - 6w = 101\]Show All Steps Hide All Steps
Start SolutionFirst, we need to get the quadratic equation in standard form. This is,
\[9{w^2} - 6w - 101 = 0\] Show Step 2Now we need to identify the values for the quadratic formula.
\[a = 9\hspace{0.25in}b = - 6\hspace{0.25in}c = - 101\] Show Step 3Plugging these into the quadratic formula gives,
\[\begin{align*}w = \frac{{ - \left( { - 6} \right) \pm \sqrt {{{\left( { - 6} \right)}^2} - 4\left( 9 \right)\left( { - 101} \right)} }}{{2\left( 9 \right)}} = \frac{{6 \pm \sqrt {3672} }}{{18}} & = \frac{{6 \pm \sqrt {\left( {36} \right)\left( {102} \right)} }}{{18}}\\ & = \frac{{6 \pm 6\sqrt {102} }}{{18}} = \frac{{1 \pm \sqrt {102} }}{3}\end{align*}\]The two solutions to this equation are then : \[\require{bbox} \bbox[2pt,border:1px solid black]{{w = \frac{1}{3} - \frac{{\sqrt {102} }}{3}\,\,{\mbox{and }}w = \frac{1}{3} + \frac{{\sqrt {102} }}{3}}}\] .