# Use Vieta's formulas to find the sum and product of the roots. x^2+3x-5=0

Question
Use Vieta's formulas to find the sum and product of the roots.
$$\displaystyle{x}^{{2}}+{3}{x}-{5}={0}$$

2021-02-07
Compare with $$\displaystyle{a}{x}^{{2}}+{b}{x}+{c}={0}$$
$$\displaystyle\Rightarrow{a}={1},{b}={3},{c}=-{5}$$
sum of the roots $$\displaystyle=-\frac{{b}}{{a}}=-\frac{{3}}{{1}}=-{3}$$
product of the roots $$\displaystyle=\frac{{c}}{{a}}=-\frac{{5}}{{1}}=-{5}$$

### Relevant Questions

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.
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) $$\displaystyle{f{{\left({x}\right)}}}{6}{x}^{{2}}+{7}{x}-{3}$$
Find the roots of the function
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Use the Quadratic Formula to solve $$\displaystyle{8}{x}^{{2}}−{24}{x}+{18}={0}.$$
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Solve the following quadratic equation $$\displaystyle{3}{x}^{{2}}+{5}{x}+{2}={0}$$
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.
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.
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
b) $$\displaystyle\lim{x}_{{{x}\to\infty}}{f{{\left({x}\right)}}}$$