We have mentioned before that the
roots of a quadratic equation are the solutions or answers you get from solving the quadatic equation. Working back from the answers, will take you to an equation.
Find an equation with roots 13 and -5
The step before giving the solutions would be:
$$(x-13)(x+5)=0$$
Notice that the signs in the brackets are opposite of the given roots.
$${x}^{2}-8x-65=0$$
Of course, there would be other possibilities as well when each term on each side of the
equal to sign is multiplied by a constant.
Find an equation with roots
$-\frac{3}{2}$ and 4
Notice that if
$x=-\frac{3}{2}$ then
$2x+3=0$
Therefore the two brackets will be:
$$(2x+3)(x-4)=0$$
The equation is:
$$2{x}^{2}-5x-12=0$$
Theory of quadratic equations - advanced
This section is not in the syllabus, but it gives one a good understanding about some of the solutions of the quadratic equations.
What is the discriminant of a quadratic equation?
Consider a general quadratic function of the form
$f\left(x\right)=a{x}^{2}+bx+c$ . The
discriminant is defined as:
$$\Delta ={b}^{2}-4ac.$$
This is the expression under the square root in the formula for the roots of this function. We have already seen that whether the roots exist or not depends on whether this factor
$\Delta $ is negative or positive.
The nature of the roots
Real roots (
$\Delta \ge 0$ )
Consider
$\Delta \ge 0$ for some quadratic function
$f\left(x\right)=a{x}^{2}+bx+c$ . In this case there are solutions to the equation
$f\left(x\right)=0$ given
by the formula
If the expression under the square root is non-negative then the square root exists. These are the roots of the function
$f\left(x\right)$ .
There various possibilities are summarised in the figure below.
Equal roots (
$\Delta =0$ )
If
$\Delta =0$ , then the roots are equal and, from the formula, these
are given by
$$x=-\frac{b}{2a}$$
Unequal roots (
$\Delta >0$ )
There will be 2 unequal roots if
$\Delta >0$ . The roots of
$f\left(x\right)$ are
rational if
$\Delta $ is a perfect square (a number which is the square of a rational number), since, in this case,
$\sqrt{\Delta}$ is rational. Otherwise, if
$\Delta $ is not a perfect square, then the roots are
irrational .
Imaginary roots (
$\Delta <0$ )
If
$\Delta <0$ , then the solution to
$f\left(x\right)=a{x}^{2}+bx+c=0$ contains the square root of a negative number and therefore there are no real solutions. We therefore say that the roots of
$f\left(x\right)$ are
imaginary (the graph of the function
$f\left(x\right)$ does not intersect the
$x$ -axis).
If
$b=0$ , discuss the nature of the roots of the equation.
If
$b=2$ , find the value(s) of
$k$ for which the roots are equal.
[IEB, Nov. 2002, HG] Show that
${k}^{2}{x}^{2}+2=kx-{x}^{2}$ has non-real roots for all real values for
$k$ .
[IEB, Nov. 2003, HG] The equation
${x}^{2}+12x=3k{x}^{2}+2$ has real roots.
Find the largest integral value of
$k$ .
Find one rational value of
$k$ , for which the above equation has rational roots.
[IEB, Nov. 2003, HG] In the quadratic equation
$p{x}^{2}+qx+r=0$ ,
$p$ ,
$q$ and
$r$ are positive real numbers and form a geometric sequence. Discuss the nature of the roots.
Find a value of
$k$ for which the roots are equal.
Find an integer
$k$ for which the roots of the equation will be rational and unequal.
[IEB, Nov. 2005, HG]
Prove that the roots of the equation
${x}^{2}-(a+b)x+ab-{p}^{2}=0$ are real for all real values of
$a$ ,
$b$ and
$p$ .
When will the roots of the equation be equal?
[IEB, Nov. 2005, HG] If
$b$ and
$c$ can take on only the values 1, 2 or 3, determine all pairs (
$b;\phantom{\rule{0.222222em}{0ex}}c$ ) such that
${x}^{2}+bx+c=0$ has real roots.
End of chapter exercises
Solve:
${x}^{2}-x-1=0$ (Give your answer correct to two decimal places.)
Solve:
$16(x+1)={x}^{2}(x+1)$
Solve:
${y}^{2}+3+{\displaystyle \frac{12}{{y}^{2}+3}}=7$ (Hint: Let
${y}^{2}+3=k$ and solve for
$k$ first and use the answer to solve
$y$ .)
Solve for
$x$ :
$2{x}^{4}-5{x}^{2}-12=0$
Solve for
$x$ :
$x(x-9)+14=0$
${x}^{2}-x=3$ (Show your answer correct to ONE decimal place.)
$x+2={\displaystyle \frac{6}{x}}$ (correct to 2 decimal places)
$\frac{1}{x+1}}+{\displaystyle \frac{2x}{x-1}}=1$
Solve for
$x$ by completing the square:
${x}^{2}-px-4=0$
The equation
$a{x}^{2}+bx+c=0$ has roots
$x={\textstyle \frac{2}{3}}$ and
$x=-4$ . Find one set of possible values for
$a$ ,
$b$ and
$c$ .
The two roots of the equation
$4{x}^{2}+px-9=0$ differ by 5. Calculate the value of
$p$ .
An equation of the form
${x}^{2}+bx+c=0$ is written
on the board. Saskia and Sven copy it down incorrectly. Saskia hasa mistake in the constant term and obtains the solutions -4 and 2.
Sven has a mistake in the coefficient of
$x$ and obtains the solutions
1 and -15. Determine the correct equation that was on theboard.
Bjorn stumbled across the following formula to solve
the quadratic equation
$a{x}^{2}+bx+c=0$ in a foreign textbook.
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