Question

find the value of beta by a alpha + b + alpha by a beta + b,where alpha + beta is equal to minus b by a and Alpha into beta is equals to C by a.​

Answer 1

Answer:

$\mathbf{\color{purple}{\huge{✮}}}\mathbf{\blue{\huge{A}}}\mathbf{\color{aqua}{\huge{N}}}\mathbf{\green{\huge{S}}}\mathbf{\color{yellow}{\huge{W}}}\mathbf{\orange{\huge{E}}}\mathbf{\red{\huge{R}}}\mathbf{\color{maroon}{\huge{✮}}}$

oots of a quadratic equation

We learned on the previous page (The Quadratic Formula), in general there are two roots for any quadratic equation \displaystyle{a}{x}^{2}+{b}{x}+{c}={0}ax

2

+bx+c=0. Let's denote those roots \displaystyle\alphaα and \displaystyle\betaβ, as follows:

\displaystyle\alpha=\frac{{-{b}+\sqrt{{{b}^{2}-{4}{a}{c}}}}}{{{2}{a}}}α=

2a

−b+

b

2

−4ac

and

\displaystyle\beta=\frac{{-{b}-\sqrt{{{b}^{2}-{4}{a}{c}}}}}{{{2}{a}}}β=

2a

−b−

b

2

−4ac

Sum of the roots α and β

We can add \displaystyle\alphaα and \displaystyle\betaβ as follows:

\displaystyle\alpha+\beta=\frac{{-{b}+\sqrt{{{b}^{2}-{4}{a}{c}}}}}{{{2}{a}}}+\frac{{-{b}-\sqrt{{{b}^{2}-{4}{a}{c}}}}}{{{2}{a}}}α+β=

2a

−b+

b

2

−4ac

+

2a

−b−

b

2

−4ac

\displaystyle=\frac{{-{2}{b}+{0}}}{{{2}{a}}}=

2a

−2b+0

\displaystyle=-\frac{b}{{a}}=−

a

b

Product of the roots α and β

We can multiply \displaystyle\alphaα and \displaystyle\betaβ as follows. First, recall that in general,

\displaystyle{\left({X}+{Y}\right)}{\left({X}-{Y}\right)}={X}^{2}-{Y}^{2}(X+Y)(X−Y)=X 2 −Y 2 and

\displaystyle{\left(\sqrt{{{X}}}\right)}^{2}={X}( X)2 =X

We make use of these to obtain:

\displaystyle\alpha\times\beta=\frac{{-{b}+\sqrt{{{b}^{2}-{4}{a}{c}}}}}{{{2}{a}}}\times\frac{{-{b}-\sqrt{{{b}^{2}-{4}{a}{c}}}}}{{{2}{a}}}α×β= 2a−b+ b 2 −4ac × 2a−b− b 2 −4ac

\displaystyle=\frac{{{\left(-{b}\right)}^{2}-{\left(\sqrt{{{b}^{2}-{4}{a}{c}}}\right)}^{2}}}{{\left({2}{a}\right)}^{2}}= (2a) 2(−b) 2 −( b 2 −4ac ) 2

\displaystyle=\frac{{{b}^{2}-{\left({b}^{2}-{4}{a}{c}\right)}}}{{{4}{a}^{2}}}= 4a 2b2 −(b 2 −4ac)

\displaystyle=\frac{{{4}{a}{c}}}{{{4}{a}^{2}}}= 4a24ac

\displaystyle=\frac{c}{{a}}= ac

Summary

The sum of the roots \displaystyle\alphaα and \displaystyle\betaβ of a quadratic equation are:

\displaystyle\alpha+\beta=-\frac{b}{{a}}α+β=−ab

The product of the roots \displaystyle\alphaα and \displaystyle\betaβ is given by:

\displaystyle\alpha\beta=\frac{c}{{a}}αβ= ac

It's also important to realize that if \displaystyle\alphaα and \displaystyle\betaβ are roots, then:

\displaystyle{\left({x}-\alpha\right)}{\left({x}-\beta\right)}={0}(x−α)(x−β)=0

We can expand the left side of the above equation to give us the following form for the quadratic formula:

\displaystyle{x}^{2}-{\left(\alpha+\beta\right)}{x}\alpha\beta={0}x 2

−(α+β)x+αβ=0

Let's use these results to solve a few problems.