Factoring is used to solve quadratics of the form ax^2+bx+c=0 when th eroots are rational. Find the roots of the following quadratic functions by factoring: a) f(x)=x^2-5x-14 b) f(x)=x^2-64 c) f(x) 6x^2+7x-3

Question
Quadratics
asked 2021-02-11
Factoring is used to solve quadratics of the form \(\displaystyle{a}{x}^{{2}}+{b}{x}+{c}={0}\) when th eroots are rational. Find the roots of the following quadratic functions by factoring:
a) \(\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}-{5}{x}-{14}\)
b) \(\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}-{64}\)
c) \(\displaystyle{f{{\left({x}\right)}}}{6}{x}^{{2}}+{7}{x}-{3}\)

Answers (1)

2021-02-12
a) \(\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}-{5}{x}-{14}\)
\(\displaystyle{f{{\left({x}\right)}}}\Rightarrow{x}^{{2}}-{5}{x}-{14}={0}\)
\(\displaystyle{x}^{{2}}-{7}{x}+{2}{x}-{14}={0}\)
\(\displaystyle{x}{\left({x}-{7}\right)}+{2}{\left({x}-{7}\right)}={0}\)
\(\displaystyle{\left({x}+{2}\right)}{\left({x}-{7}\right)}={0}\)
Hence, the roots are x=-2,7ZSK
b) \(\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}-{64}\)
\(\displaystyle{f{{\left({x}\right)}}}={0}\Rightarrow{x}^{{2}}-{64}={0}\)
\(\displaystyle{\left({x}-{8}\right)}{\left({x}+{8}\right)}={0}\)
Hence, the roots are x=-8,8
c) \(\displaystyle{f{{\left({x}\right)}}}{6}{x}^{{2}}+{7}{x}-{3}\)
\(\displaystyle{f{{\left({x}\right)}}}={0}\Rightarrow{6}{x}^{{2}}+{7}{x}-{3}={0}\)
\(\displaystyle{6}{x}^{{2}}+{9}{x}-{2}{x}-{3}={0}\)
\(\displaystyle{3}{x}{\left({2}{x}+{3}\right)}-{1}{\left({2}{x}+{3}\right)}={0}\)
\(\displaystyle{\left({3}{x}-{1}\right)}{\left({2}{x}+{3}\right)}={0}\)
Hence, the roots are \(\displaystyle{x}=-\frac{{1}}{{3}},-\frac{{3}}{{2}}\)
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