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

### Relevant Questions

Which of the following is/are always false ?
(a) A quadratic equation with rational coefficients has zero or two irrational roots.
(b) A quadratic equation with real coefficients has zero or two non - real roots
(c) A quadratic equation with irrational coefficients has zero or two rational roots.
(d) A quadratic equation with integer coefficients has zero or two irrational roots.
Consider the polynomial $$\displaystyle{f{{\left({X}\right)}}}={x}^{{4}}+{1}$$
a) Explain why f has no real roots, and why this means f mustfactor as a product of two ireducible quadratics.
b) Factor f and find all of its complex roots.
Use a system of linear equations to find the quadratic function
$$f(x) = ax^22+bx+c$$
that satisfies the given conditions. Solve the system using matrices.
f(-2) = 6, f(1) = -3, f(2) = -14
f(x) =?
The factored form of a quadratic function is f(x)=(x‒p)(x‒q). The standard form of a quadratic function is $$\displaystyle{f{{\left({x}\right)}}}={a}{x}^{{2}}+{b}{x}+{c}$$. The factored form for a quadratic function tells us the x-intercepts of the quadratic function, while the standard form for a quadratic function tells us the y-intercept of the quadratic function.
If p=2 and q=‒3, use what you have learned about multiplying polynomials to
- write the factored form of the quadratic function f(x)
- write the standard form for the quadratic function f(x)
A function is a ratio of quadratic functions and has a vertical asymptote x =4 and just one -intercept, x =1. It is known f that has a removable discontinuity at x =- 1 and $$\displaystyle\lim_{{{x}\to{1}}}{f{{\left({x}\right)}}}={2}$$. Evaluate
a) f(0)
b) $$\displaystyle\lim{x}_{{{x}\to\infty}}{f{{\left({x}\right)}}}$$
Solve the following quadratic equation $$\displaystyle{6}{x}^{{2}}-{x}-{15}={0}$$
Find the vertex of the qudratic function $$\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}-{6}{x}+{42}$$, then express the qudratic function in standart form $$\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}-{h}\right)}^{{2}}+{k}$$ and state whether the vertex is a minimum or maximum. Enter exat answers only, no approximations.
Standard transformations can be used to help graph rational functions of the form $$\displaystyle{f{{\left({x}\right)}}}={\frac{{{A}}}{{{B}{x}-{C}}}}+{D}$$ Explain how the parameters A, B, C, and D relate to the graph of the rational functions.
Quadratics by factoring $$\displaystyle{2}{z}{\left({5}{z}-{2}\right)}=-{5}{z}+{2}$$