Chapter 4.5, Problem 9A

Solution

(a.)

To Explain: A quadratic function or equation in standard form can be written if the roots are known.

It has been explained how a quadratic function or equation in standard form can be written if the roots are known.

Given:

The roots of a quadratic equation.

Concept used:

There is a relation between the roots of a quadratic equation and its coefficients.

Calculation:

Let the roots of a quadratic equation be $a$ and $b$ .

Then, the quadratic equation can be written as $\left(x-a\right)\left(x-b\right)=0$ .

Simplifying,

  $x\left(x-b\right)-a\left(x-b\right)=0$

On further simplification,

  $x^2-bx-ax+ab=0$

Finally,

  $x^2-\left(a+b\right)x+ab=0$

So, the quadratic equation, whose roots are $a$ and $b$ , can be written in standard form as $x^2-\left(a+b\right)x+ab=0$ .

Conclusion:

It has been explained how a quadratic function or equation in standard form can be written if the roots are known.

(b.)

To Demonstrate: The method for writing the quadratic equation in standard form for each pair of roots given.

The method for writing the quadratic equation in standard form for each pair of roots given has been demonstrated.

Given:

The following pairs of roots:

  $5$ and $3$ ,

  $2.5$ and $4$ ,

  $-4$ and $4$ ,

  $5$ and $10$ ,

  $\frac{3}{2}$ and $-2$

Concept used:

There is a relation between the roots of a quadratic equation and its coefficients.

Calculation:

As determined previously, the quadratic equation, whose roots are $a$ and $b$ , can be written in standard form as $x^2-\left(a+b\right)x+ab=0$ .

The first pair of roots is $5$ and $3$ .

So, $a=5$ and $b=3$ .

Put these values in $x^2-\left(a+b\right)x+ab=0$ to get,

  $x^2-\left(5+3\right)x+\left(5\right)\left(3\right)=0$

Simplifying,

  $x^2-8x+15=0$

This is the required quadratic equation.

The second pair of roots is $2.5$ and $4$ .

So, $a=2.5$ and $b=4$ .

Put these values in $x^2-\left(a+b\right)x+ab=0$ to get,

  $x^2-\left(2.5+4\right)x+\left(2.5\right)\left(4\right)=0$

Simplifying,

  $x^2-6.5x+10=0$

On further simplification,

  $10x^2-65x+100=0$

Finally,

  $2x^2-13x+20=0$

This is the required quadratic equation.

The third pair of roots is $-4$ and $4$ .

So, $a=-4$ and $b=4$ .

Put these values in $x^2-\left(a+b\right)x+ab=0$ to get,

  $x^2-\left(-4+4\right)x+\left(-4\right)\left(4\right)=0$

Simplifying,

  $x^2-16=0$

This is the required quadratic equation.

The fourth pair of roots is $5$ and $10$ .

So, $a=5$ and $b=10$ .

Put these values in $x^2-\left(a+b\right)x+ab=0$ to get,

  $x^2-\left(5+10\right)x+\left(5\right)\left(10\right)=0$

Simplifying,

  $x^2-15x+50=0$

This is the required quadratic equation.

The fifth pair of roots is $\frac{3}{2}$ and $-2$ .

So, $a=\frac{3}{2}$ and $b=-2$ .

Put these values in $x^2-\left(a+b\right)x+ab=0$ to get,

  $x^2-\left(\frac{3}{2}-2\right)x+\left(\frac{3}{2}\right)\left(-2\right)=0$

Simplifying,

  $x^2-\left(-\frac{1}{2}\right)x+\left(3\right)\left(-1\right)=0$

On further simplification,

  $x^2+\frac{1}{2}x-3=0$

Finally,

  $2x^2+x-6=0$

This is the required quadratic equation.

Conclusion:

The method for writing the quadratic equation in standard form for each pair of roots given has been demonstrated.