Solved on Jan 16, 2024

Find the quadratic equation with two $x$ -intercepts from the given options: $-2 x^{2}-5=0$ , $x^{2}-4 x+4=0$ , $-x^{2}-x=0$ , $x^{2}+12=0$ , $-x^{2}-12=0$ .

STEP 1

Assumptions
1. A quadratic equation has the form $ax^2 + bx + c = 0$ .
2. A quadratic equation has two $x$ -intercepts if and only if it has two distinct real solutions.
3. The discriminant of a quadratic equation, given by $D = b^2 - 4ac$ , determines the nature of the roots: - If $D > 0$ , the equation has two distinct real solutions (and thus two $x$ -intercepts). - If $D = 0$ , the equation has one real solution (a repeated root, thus one $x$ -intercept). - If $D < 0$ , the equation has no real solutions (and thus no $x$ -intercepts).

STEP 2

We will calculate the discriminant for each given quadratic equation to determine which one has two $x$ -intercepts.

STEP 3

For equation b, $-2x^2 - 5 = 0$ , the coefficients are $a = -2$ , $b = 0$ , and $c = -5$ .

STEP 4

Calculate the discriminant for equation b.
$D_b = b^2 - 4ac = 0^2 - 4(-2)(-5) = -40$

STEP 5

Since $D_b < 0$ , equation b does not have two $x$ -intercepts.

STEP 6

For equation c, $x^2 - 4x + 4 = 0$ , the coefficients are $a = 1$ , $b = -4$ , and $c = 4$ .

STEP 7

Calculate the discriminant for equation c.
$D_c = b^2 - 4ac = (-4)^2 - 4(1)(4) = 16 - 16 = 0$

STEP 8

Since $D_c = 0$ , equation c does not have two $x$ -intercepts.

STEP 9

For equation d, $-x^2 - x = 0$ , the coefficients are $a = -1$ , $b = -1$ , and $c = 0$ .

STEP 10

Calculate the discriminant for equation d.
$D_d = b^2 - 4ac = (-1)^2 - 4(-1)(0) = 1$

STEP 11

Since $D_d > 0$ , equation d has two $x$ -intercepts.

STEP 12

For equation e, $x^2 + 12 = 0$ , the coefficients are $a = 1$ , $b = 0$ , and $c = 12$ .

STEP 13

Calculate the discriminant for equation e.
$D_e = b^2 - 4ac = 0^2 - 4(1)(12) = -48$

STEP 14

Since $D_e < 0$ , equation e does not have two $x$ -intercepts.

STEP 15

For equation f, $-x^2 - 12 = 0$ , the coefficients are $a = -1$ , $b = 0$ , and $c = -12$ .

STEP 16

Calculate the discriminant for equation f.
$D_f = b^2 - 4ac = 0^2 - 4(-1)(-12) = -48$

STEP 17

Since $D_f < 0$ , equation f does not have two $x$ -intercepts.

STEP 18

Based on the discriminant calculations, the only equation with two $x$ -intercepts is equation d, $-x^2 - x = 0$ .
The correct answer is d. $-x^2 - x = 0$ .

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Find the quadratic equation with two $x$ -intercepts from the given options: $-2 x^{2}-5=0$ , $x^{2}-4 x+4=0$ , $-x^{2}-x=0$ , $x^{2}+12=0$ , $-x^{2}-12=0$ .

STEP 1

STEP 2

We will calculate the discriminant for each given quadratic equation to determine which one has two $x$ -intercepts.

STEP 3

For equation b, $-2x^2 - 5 = 0$ , the coefficients are $a = -2$ , $b = 0$ , and $c = -5$ .

STEP 4

Calculate the discriminant for equation b.
$D_b = b^2 - 4ac = 0^2 - 4(-2)(-5) = -40$

STEP 5

Since $D_b < 0$ , equation b does not have two $x$ -intercepts.

STEP 6

For equation c, $x^2 - 4x + 4 = 0$ , the coefficients are $a = 1$ , $b = -4$ , and $c = 4$ .

STEP 7

Calculate the discriminant for equation c.
$D_c = b^2 - 4ac = (-4)^2 - 4(1)(4) = 16 - 16 = 0$

STEP 8

Since $D_c = 0$ , equation c does not have two $x$ -intercepts.

STEP 9

For equation d, $-x^2 - x = 0$ , the coefficients are $a = -1$ , $b = -1$ , and $c = 0$ .

STEP 10

Calculate the discriminant for equation d.
$D_d = b^2 - 4ac = (-1)^2 - 4(-1)(0) = 1$

STEP 11

Since $D_d > 0$ , equation d has two $x$ -intercepts.

STEP 12

For equation e, $x^2 + 12 = 0$ , the coefficients are $a = 1$ , $b = 0$ , and $c = 12$ .

STEP 13

Calculate the discriminant for equation e.
$D_e = b^2 - 4ac = 0^2 - 4(1)(12) = -48$

STEP 14

Since $D_e < 0$ , equation e does not have two $x$ -intercepts.

STEP 15

For equation f, $-x^2 - 12 = 0$ , the coefficients are $a = -1$ , $b = 0$ , and $c = -12$ .

STEP 16

Calculate the discriminant for equation f.
$D_f = b^2 - 4ac = 0^2 - 4(-1)(-12) = -48$

STEP 17

Since $D_f < 0$ , equation f does not have two $x$ -intercepts.

STEP 18

Based on the discriminant calculations, the only equation with two $x$ -intercepts is equation d, $-x^2 - x = 0$ .
The correct answer is d. $-x^2 - x = 0$ .

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Find the quadratic equation with two xxx-intercepts from the given options: −2x2−5=0-2 x^{2}-5=0−2x2−5=0, x2−4x+4=0x^{2}-4 x+4=0x2−4x+4=0, −x2−x=0-x^{2}-x=0−x2−x=0, x2+12=0x^{2}+12=0x2+12=0, −x2−12=0-x^{2}-12=0−x2−12=0.

STEP 1

STEP 2

We will calculate the discriminant for each given quadratic equation to determine which one has two xxx-intercepts.

STEP 3

For equation b, −2x2−5=0-2x^2 - 5 = 0−2x2−5=0, the coefficients are a=−2a = -2a=−2, b=0b = 0b=0, and c=−5c = -5c=−5.

STEP 4

Calculate the discriminant for equation b.Db=b2−4ac=02−4(−2)(−5)=−40D_b = b^2 - 4ac = 0^2 - 4(-2)(-5) = -40Db​=b2−4ac=02−4(−2)(−5)=−40

STEP 5

Since Db<0D_b < 0Db​<0, equation b does not have two xxx-intercepts.

STEP 6

For equation c, x2−4x+4=0x^2 - 4x + 4 = 0x2−4x+4=0, the coefficients are a=1a = 1a=1, b=−4b = -4b=−4, and c=4c = 4c=4.

STEP 7

Calculate the discriminant for equation c.Dc=b2−4ac=(−4)2−4(1)(4)=16−16=0D_c = b^2 - 4ac = (-4)^2 - 4(1)(4) = 16 - 16 = 0Dc​=b2−4ac=(−4)2−4(1)(4)=16−16=0

STEP 8

Since Dc=0D_c = 0Dc​=0, equation c does not have two xxx-intercepts.

STEP 9

For equation d, −x2−x=0-x^2 - x = 0−x2−x=0, the coefficients are a=−1a = -1a=−1, b=−1b = -1b=−1, and c=0c = 0c=0.

STEP 10

Calculate the discriminant for equation d.Dd=b2−4ac=(−1)2−4(−1)(0)=1D_d = b^2 - 4ac = (-1)^2 - 4(-1)(0) = 1Dd​=b2−4ac=(−1)2−4(−1)(0)=1

STEP 11

Since Dd>0D_d > 0Dd​>0, equation d has two xxx-intercepts.

STEP 12

For equation e, x2+12=0x^2 + 12 = 0x2+12=0, the coefficients are a=1a = 1a=1, b=0b = 0b=0, and c=12c = 12c=12.

STEP 13

Calculate the discriminant for equation e.De=b2−4ac=02−4(1)(12)=−48D_e = b^2 - 4ac = 0^2 - 4(1)(12) = -48De​=b2−4ac=02−4(1)(12)=−48

STEP 14

Since De<0D_e < 0De​<0, equation e does not have two xxx-intercepts.

STEP 15

For equation f, −x2−12=0-x^2 - 12 = 0−x2−12=0, the coefficients are a=−1a = -1a=−1, b=0b = 0b=0, and c=−12c = -12c=−12.

STEP 16

Calculate the discriminant for equation f.Df=b2−4ac=02−4(−1)(−12)=−48D_f = b^2 - 4ac = 0^2 - 4(-1)(-12) = -48Df​=b2−4ac=02−4(−1)(−12)=−48

STEP 17

Since Df<0D_f < 0Df​<0, equation f does not have two xxx-intercepts.

STEP 18

Based on the discriminant calculations, the only equation with two xxx-intercepts is equation d, −x2−x=0-x^2 - x = 0−x2−x=0.The correct answer is d. −x2−x=0-x^2 - x = 0−x2−x=0.

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Find the quadratic equation with two $x$ -intercepts from the given options: $-2 x^{2}-5=0$ , $x^{2}-4 x+4=0$ , $-x^{2}-x=0$ , $x^{2}+12=0$ , $-x^{2}-12=0$ .

We will calculate the discriminant for each given quadratic equation to determine which one has two $x$ -intercepts.

For equation b, $-2x^2 - 5 = 0$ , the coefficients are $a = -2$ , $b = 0$ , and $c = -5$ .

Calculate the discriminant for equation b.
$D_b = b^2 - 4ac = 0^2 - 4(-2)(-5) = -40$

Since $D_b < 0$ , equation b does not have two $x$ -intercepts.

For equation c, $x^2 - 4x + 4 = 0$ , the coefficients are $a = 1$ , $b = -4$ , and $c = 4$ .

Calculate the discriminant for equation c.
$D_c = b^2 - 4ac = (-4)^2 - 4(1)(4) = 16 - 16 = 0$

Since $D_c = 0$ , equation c does not have two $x$ -intercepts.

For equation d, $-x^2 - x = 0$ , the coefficients are $a = -1$ , $b = -1$ , and $c = 0$ .

Calculate the discriminant for equation d.
$D_d = b^2 - 4ac = (-1)^2 - 4(-1)(0) = 1$

Since $D_d > 0$ , equation d has two $x$ -intercepts.

For equation e, $x^2 + 12 = 0$ , the coefficients are $a = 1$ , $b = 0$ , and $c = 12$ .

Calculate the discriminant for equation e.
$D_e = b^2 - 4ac = 0^2 - 4(1)(12) = -48$

Since $D_e < 0$ , equation e does not have two $x$ -intercepts.

For equation f, $-x^2 - 12 = 0$ , the coefficients are $a = -1$ , $b = 0$ , and $c = -12$ .

Calculate the discriminant for equation f.
$D_f = b^2 - 4ac = 0^2 - 4(-1)(-12) = -48$

Since $D_f < 0$ , equation f does not have two $x$ -intercepts.

Based on the discriminant calculations, the only equation with two $x$ -intercepts is equation d, $-x^2 - x = 0$ .
The correct answer is d. $-x^2 - x = 0$ .