Applications of the Discriminant 1

Question 1 of 7

Using the discriminant, find the nature of the roots of the function:

$5x^2-6x+4=0$

1. One real root
2. Two real roots
3. No real roots

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Nature of the Roots	Discriminant ( $Delta$ )
Two real roots	$Delta$ $>$ $0$
One real root	$Delta=0$
No real roots	$Delta$ $<$ $0$

Discriminant Formula

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

First, compute for the discriminant

$5x^2-6x+4=0$

$a=5$ $b=-6$ $c=4$

$Delta$	$=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$	${\color{#9a00c7}{(-6)}}^2-4\color{#00880A}{(5)}\color{#007DDC}{(4)}$	Substitute values
	$=$	$36-80$
	$=$	$-44$

This is a negative value, which means

$Delta$

$<$

$0$

Therefore, the function has No real roots

No real roots

Question 2 of 7

Identify which values of

$k$ will make the function below have one real root

$x^2-(k-8)x+4=0$

1. $k=4,12$
2. $k=-4,2$
3. $k=0$
4. $k=-12,-4$

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Nature of the Roots	Discriminant ( $Delta$ )
Two real roots	$Delta$ $>$ $0$
One real root	$Delta=0$
No real roots	$Delta$ $<$ $0$

Discriminant Formula

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

First, compute for the discriminant

$x^2-(k-8)x+4=0$

$a=1$ $b=k-8$ $c=4$

$Delta$	$=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$	${\color{#9a00c7}{(k-8)}}^2-4\color{#00880A}{(1)}\color{#007DDC}{(4)}$	Substitute values
	$=$	$k^2-16k+64-16$
	$=$	$k^2-16k+48$

Remember that for a function to have one real root,

$Delta$

$=$

$0$

Substitute the

$Delta$ computed previously, and then solve for

$k$

$Delta$	$=$	$0$
$k^2-16k+48$	$=$	$0$

[insert cross method with two

$k$ ’s on the right and

$-4$ &

$-12$ on the left]

$(k-4)(k-12)$	$=$	$0$
$k=4,12$

$k=4,12$

Question 3 of 7

Identify which values of

$k$ will make the function below have two real roots

$x^2-3kx+9=0$

1. $k$ $<$ $-2$ and $k$ $>$ $2$
2. $k$ $<$ $-3$ and $k$ $>$ $3$
3. $-2$ $<$ $k$ $<$ $2$
4. $-3$ $<$ $k$ $<$ $3$

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Nature of the Roots	Discriminant ( $Delta$ )
Two real roots	$Delta$ $>$ $0$
One real root	$Delta=0$
No real roots	$Delta$ $<$ $0$

Discriminant Formula

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

First, compute for the discriminant

$x^2-3kx+9=0$

$a=1$ $b=-3k$ $c=9$

$Delta$	$=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$	${\color{#9a00c7}{(-3k)}}^2-4\color{#00880A}{(1)}\color{#007DDC}{(9)}$	Substitute values
	$=$	$9k^2-36$

Remember that for a function to have two real roots,

$Delta$

$>$

$0$

Substitute the

$Delta$ computed previously, and then solve for

$k$

$Delta$	$>$	$0$
$9k^2-36$	$>$	$0$
$9(k^2-4)$	$>$	$0$
$9(k-2)(k+2)$	$>$	$0$
$k=2$		$k=-2$

To determine which region around

$k=-2$ and

$k=2$ would be included, plot these points and make a rough sketch of

$9k^2-36$

Replace the

$x$ axis with

$k$ axis and draw an upward parabola since

$9$ is positive

Remember that

$Delta$ must be positive

Therefore,

$k$

$<$

$-2$ and

$k$

$>$

$2$

$k$

$<$

$-2$ and

$k$

$>$

$2$

Question 4 of 7

Identify which values of

$k$ will make the function below have real roots

$x^2+(k+2)x+4=0$

1. $-2$ $≤$ $k$ $≤$ $6$
2. $-6$ $≤$ $k$ $≤$ $2$
3. $k$ $≤$ $-2$ and $k$ $≥$ $6$
4. $k$ $≤$ $-6$ and $k$ $≥$ $2$

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Nature of the Roots	Discriminant ( $Delta$ )
Two real roots	$Delta$ $>$ $0$
One real root	$Delta=0$
No real roots	$Delta$ $<$ $0$

Discriminant Formula

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

First, compute for the discriminant

$x^2+(k+2)x+4=0$

$a=1$ $b=k+2$ $c=4$

$Delta$	$=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$	${\color{#9a00c7}{(k+2)}}^2-4\color{#00880A}{(1)}\color{#007DDC}{(4)}$	Substitute values
	$=$	$k^2+4k+4-16$
	$=$	$k^2+4k-12$

A function that has real roots can have either one or two real roots, hence

$Delta$

$≥$

$0$

Substitute the

$Delta$ computed previously, and then solve for

$k$

$Delta$	$≥$	$0$
$k^2+4k-12$	$≥$	$0$

$(k+6)(k-2)$	$≥$	$0$
$k=-6$		$k=2$

To determine which region around

$k=-6$ and

$k=2$ would be included, plot these points and make a rough sketch of

$k^2+4k-12$

Replace the

$x$ axis with

$k$ axis and draw an upward parabola since

$1$ is positive

Remember that

$Delta$ must be positive

Therefore,

$k$

$≤$

$-6$ and

$k$

$≥$

$2$

$k$

$≤$

$-6$ and

$k$

$≥$

$2$

Question 5 of 7

Identify which values of

$k$ will make the function below have one real root

$kx^2-4x+k=0$

1. $k=0,2$
2. $k=2$
3. $k=-2$
4. $k=-2,2$

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Nature of the Roots	Discriminant ( $Delta$ )
Two real roots	$Delta$ $>$ $0$
One real root	$Delta=0$
No real roots	$Delta$ $<$ $0$

Discriminant Formula

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

First, compute for the discriminant

$kx^2-4x+k=0$

$a=k$ $b=-4$ $c=k$

$Delta$	$=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$	${\color{#9a00c7}{(-4)}}^2-4\color{#00880A}{(k)}\color{#007DDC}{(k)}$	Substitute values
	$=$	$16-4k^2$

Remember that the function must have one real root, hence

$Delta=0$

Substitute the

$Delta$ computed previously, and then solve for

$k$

$Delta$	$=$	$0$
$16-4k^2$	$=$	$0$
$4(4-k^2)$	$=$	$0$
$4(2-k)(2+k)$	$=$	$0$
$k=-2$		$k=2$

$k=-2,2$

Question 6 of 7

Identify which values of

$m$ will make the function below have one real root

$2x^2+mx+8=0$

1. $m=-8$
2. $m=0$
3. $m=-8,8$
4. $m=0,8$

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Nature of the Roots	Discriminant ( $Delta$ )
Two real roots	$Delta$ $>$ $0$
One real root	$Delta=0$
No real roots	$Delta$ $<$ $0$

Discriminant Formula

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

First, compute for the discriminant

$2x^2+mx+8=0$

$a=2$ $b=m$ $c=8$

$Delta$	$=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$	${\color{#9a00c7}{m}}^2-4\color{#00880A}{(2)}\color{#007DDC}{(8)}$	Substitute values
	$=$	$m^2-64$

Remember that the function must have one real root, hence

$Delta=0$

Substitute the

$Delta$ computed previously, and then solve for

$m$

$Delta$	$=$	$0$
$m^2-64$	$=$	$0$
$(m+8)(m-8)$	$=$	$0$
$m=-8,8$

$m=-8,8$

Question 7 of 7

For what value of

$k$ will the line

$y=3x-k$ be tangent to the parabola of

$y=2x^2-x+3$

$k=$ (-1)

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Nature of the Roots	Discriminant ( $Delta$ )
Two real roots	$Delta$ $>$ $0$
One real root	$Delta=0$
No real roots	$Delta$ $<$ $0$

Discriminant Formula

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

Equate the two functions and solve for

$k$ in such a way that the discriminant (

$Delta$ ) will be

$0$

First, equate the two functions to create a single equation

$y=2x^2-x+3$

$y=3x-k$

$2x^2-x+3$	$=$	$3x-k$
$2x^2-x+3-3x+k$	$=$	$0$	Move all values to the left
$2x^2-4x+3+k$	$=$	$0$

Next, compute for the discriminant

$2x^2-4x+3+k$

$a=2$ $b=-4$ $c=3+k$

$Delta$	$=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$	${\color{#9a00c7}{(-4)}}^2-4\color{#00880A}{(2)}\color{#007DDC}{(3+k)}$	Substitute values
	$=$	$16-24-8k$
	$=$	$-8-8k$

Remember that the line must be a tangent to the parabola which means they meet at one point and that the previous equation must have one root, hence

$Delta=0$

Substitute the

$Delta$ computed previously, and then solve for

$k$

$Delta$	$=$	$0$
$-8-8k$	$=$	$0$
$-8-8k$ $+8$	$=$	$0$ $+8$	Add $8$ to both sides
$-8k$	$=$	$8$
$-8k$ $divide(-8)$	$=$	$8$ $divide(-8)$	Divide both sides by $-8$
$k$	$=$	$-1$

$k=-1$

Applications of the Discriminant 1

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1. Question

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Well Done!

Discriminant Formula

2. Question

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Keep Going!

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3. Question

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Nice Job!

Discriminant Formula

4. Question

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Correct!

Discriminant Formula

5. Question

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Nice Job!

Discriminant Formula

6. Question

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Fantastic!

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7. Question

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Exceptional!

Discriminant Formula

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