Question

# 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

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) $$\displaystyle{f{{\left({x}\right)}}}{6}{x}^{{2}}+{7}{x}-{3}$$

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 $$x=-2,7$$
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 $$\displaystyle{x}=-\frac{{1}}{{3}},-\frac{{3}}{{2}}$$