Solved on Jan 22, 2024

Find the quadratic equation's roots: $y = -\frac{1}{2}x^2 - 2x + \frac{7}{10}$

STEP 1

Assumptions
1. The equation given is a quadratic equation in the form $y=ax^2+bx+c$ .
2. We are asked to find the roots of the equation, which are the values of $x$ that make $y=0$ .
3. The coefficients of the equation are $a=-\frac{1}{2}$ , $b=-2$ , and $c=\frac{7}{10}$ .

STEP 2

To find the roots of the quadratic equation, we can use the quadratic formula:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

STEP 3

First, we identify the coefficients $a$ , $b$ , and $c$ from the given equation:
$a = -\frac{1}{2}, \quad b = -2, \quad c = \frac{7}{10}$

STEP 4

Next, we calculate the discriminant, which is the part under the square root in the quadratic formula:
$\Delta = b^2 - 4ac$

STEP 5

Plug in the values for $a$ , $b$ , and $c$ to calculate the discriminant:
$\Delta = (-2)^2 - 4 \left(-\frac{1}{2}\right) \left(\frac{7}{10}\right)$

STEP 6

Simplify the discriminant calculation:
$\Delta = 4 - 4 \left(-\frac{1}{2}\right) \left(\frac{7}{10}\right)$

STEP 7

Multiply the terms inside the parentheses:
$\Delta = 4 - (-2) \left(\frac{7}{10}\right)$

STEP 8

Multiply the numbers:
$\Delta = 4 - (-\frac{14}{10})$

STEP 9

Simplify the expression by converting the subtraction of a negative to addition:
$\Delta = 4 + \frac{14}{10}$

STEP 10

Convert the whole number to a fraction with a denominator of 10 to add the fractions:
$\Delta = \frac{40}{10} + \frac{14}{10}$

STEP 11

Add the fractions:
$\Delta = \frac{40}{10} + \frac{14}{10} = \frac{54}{10}$

STEP 12

Simplify the fraction:
$\Delta = \frac{27}{5}$

STEP 13

Now that we have the discriminant, we can use the quadratic formula to find the roots. We will calculate the positive and negative square roots separately.
First, calculate the positive root using the quadratic formula:
$x_1 = \frac{-b + \sqrt{\Delta}}{2a}$

STEP 14

Plug in the values for $a$ , $b$ , and $\Delta$ to calculate the positive root:
$x_1 = \frac{-(-2) + \sqrt{\frac{27}{5}}}{2 \left(-\frac{1}{2}\right)}$

STEP 15

Simplify the numerator:
$x_1 = \frac{2 + \sqrt{\frac{27}{5}}}{-1}$

STEP 16

Calculate the square root of the fraction:
$\sqrt{\frac{27}{5}} = \frac{\sqrt{27}}{\sqrt{5}} = \frac{3\sqrt{3}}{\sqrt{5}}$

STEP 17

Plug the square root back into the formula for $x_1$ :
$x_1 = \frac{2 + \frac{3\sqrt{3}}{\sqrt{5}}}{-1}$

STEP 18

Simplify the expression by distributing the denominator:
$x_1 = -2 - \frac{3\sqrt{3}}{\sqrt{5}}$

STEP 19

Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{5}$ :
$x_1 = -2 - \frac{3\sqrt{3}\sqrt{5}}{5}$

STEP 20

Simplify the expression:
$x_1 = -2 - \frac{3\sqrt{15}}{5}$

STEP 21

Now, calculate the negative root using the quadratic formula:
$x_2 = \frac{-b - \sqrt{\Delta}}{2a}$

STEP 22

Plug in the values for $a$ , $b$ , and $\Delta$ to calculate the negative root:
$x_2 = \frac{-(-2) - \sqrt{\frac{27}{5}}}{2 \left(-\frac{1}{2}\right)}$

STEP 23

Simplify the numerator:
$x_2 = \frac{2 - \frac{3\sqrt{3}}{\sqrt{5}}}{-1}$

STEP 24

Simplify the expression by distributing the denominator:
$x_2 = -2 + \frac{3\sqrt{3}}{\sqrt{5}}$

STEP 25

Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{5}$ :
$x_2 = -2 + \frac{3\sqrt{3}\sqrt{5}}{5}$

STEP 26

Simplify the expression:
$x_2 = -2 + \frac{3\sqrt{15}}{5}$

STEP 27

The roots of the quadratic equation are:
$x_1 = -2 - \frac{3\sqrt{15}}{5} \quad \text{and} \quad x_2 = -2 + \frac{3\sqrt{15}}{5}$
These are the roots of the equation $y=-\frac{1}{2} x^{2}-2 x+\frac{7}{10}$ .

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Find the quadratic equation's roots: y=−12x2−2x+710y = -\frac{1}{2}x^2 - 2x + \frac{7}{10}y=−21​x2−2x+107​

STEP 1

STEP 2

To find the roots of the quadratic equation, we can use the quadratic formula:x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac​​

STEP 3

First, we identify the coefficients aaa, bbb, and ccc from the given equation:a=−12,b=−2,c=710a = -\frac{1}{2}, \quad b = -2, \quad c = \frac{7}{10}a=−21​,b=−2,c=107​

STEP 4

Next, we calculate the discriminant, which is the part under the square root in the quadratic formula:Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac

STEP 5

Plug in the values for aaa, bbb, and ccc to calculate the discriminant:Δ=(−2)2−4(−12)(710)\Delta = (-2)^2 - 4 \left(-\frac{1}{2}\right) \left(\frac{7}{10}\right)Δ=(−2)2−4(−21​)(107​)

STEP 6

Simplify the discriminant calculation:Δ=4−4(−12)(710)\Delta = 4 - 4 \left(-\frac{1}{2}\right) \left(\frac{7}{10}\right)Δ=4−4(−21​)(107​)

STEP 7

Multiply the terms inside the parentheses:Δ=4−(−2)(710)\Delta = 4 - (-2) \left(\frac{7}{10}\right)Δ=4−(−2)(107​)

STEP 8

Multiply the numbers:Δ=4−(−1410)\Delta = 4 - (-\frac{14}{10})Δ=4−(−1014​)

STEP 9

Simplify the expression by converting the subtraction of a negative to addition:Δ=4+1410\Delta = 4 + \frac{14}{10}Δ=4+1014​

STEP 10

Convert the whole number to a fraction with a denominator of 10 to add the fractions:Δ=4010+1410\Delta = \frac{40}{10} + \frac{14}{10}Δ=1040​+1014​

STEP 11

Add the fractions:Δ=4010+1410=5410\Delta = \frac{40}{10} + \frac{14}{10} = \frac{54}{10}Δ=1040​+1014​=1054​

STEP 12

Simplify the fraction:Δ=275\Delta = \frac{27}{5}Δ=527​

STEP 13

Now that we have the discriminant, we can use the quadratic formula to find the roots. We will calculate the positive and negative square roots separately.First, calculate the positive root using the quadratic formula:x1=−b+Δ2ax_1 = \frac{-b + \sqrt{\Delta}}{2a}x1​=2a−b+Δ​​

STEP 14

Plug in the values for aaa, bbb, and Δ\DeltaΔ to calculate the positive root:x1=−(−2)+2752(−12)x_1 = \frac{-(-2) + \sqrt{\frac{27}{5}}}{2 \left(-\frac{1}{2}\right)}x1​=2(−21​)−(−2)+527​​​

STEP 15

Simplify the numerator:x1=2+275−1x_1 = \frac{2 + \sqrt{\frac{27}{5}}}{-1}x1​=−12+527​​​

STEP 16

Calculate the square root of the fraction:275=275=335\sqrt{\frac{27}{5}} = \frac{\sqrt{27}}{\sqrt{5}} = \frac{3\sqrt{3}}{\sqrt{5}}527​​=5​27​​=5​33​​

STEP 17

Plug the square root back into the formula for x1x_1x1​:x1=2+335−1x_1 = \frac{2 + \frac{3\sqrt{3}}{\sqrt{5}}}{-1}x1​=−12+5​33​​​

STEP 18

Simplify the expression by distributing the denominator:x1=−2−335x_1 = -2 - \frac{3\sqrt{3}}{\sqrt{5}}x1​=−2−5​33​​

STEP 19

Rationalize the denominator by multiplying the numerator and denominator by 5\sqrt{5}5​:x1=−2−3355x_1 = -2 - \frac{3\sqrt{3}\sqrt{5}}{5}x1​=−2−533​5​​

STEP 20

Simplify the expression:x1=−2−3155x_1 = -2 - \frac{3\sqrt{15}}{5}x1​=−2−5315​​

STEP 21

Now, calculate the negative root using the quadratic formula:x2=−b−Δ2ax_2 = \frac{-b - \sqrt{\Delta}}{2a}x2​=2a−b−Δ​​

STEP 22

Plug in the values for aaa, bbb, and Δ\DeltaΔ to calculate the negative root:x2=−(−2)−2752(−12)x_2 = \frac{-(-2) - \sqrt{\frac{27}{5}}}{2 \left(-\frac{1}{2}\right)}x2​=2(−21​)−(−2)−527​​​

STEP 23

Simplify the numerator:x2=2−335−1x_2 = \frac{2 - \frac{3\sqrt{3}}{\sqrt{5}}}{-1}x2​=−12−5​33​​​

STEP 24

Simplify the expression by distributing the denominator:x2=−2+335x_2 = -2 + \frac{3\sqrt{3}}{\sqrt{5}}x2​=−2+5​33​​

STEP 25

Rationalize the denominator by multiplying the numerator and denominator by 5\sqrt{5}5​:x2=−2+3355x_2 = -2 + \frac{3\sqrt{3}\sqrt{5}}{5}x2​=−2+533​5​​

STEP 26

Simplify the expression:x2=−2+3155x_2 = -2 + \frac{3\sqrt{15}}{5}x2​=−2+5315​​

STEP 27

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Find the quadratic equation's roots: $y = -\frac{1}{2}x^2 - 2x + \frac{7}{10}$

To find the roots of the quadratic equation, we can use the quadratic formula:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

First, we identify the coefficients $a$ , $b$ , and $c$ from the given equation:
$a = -\frac{1}{2}, \quad b = -2, \quad c = \frac{7}{10}$

Next, we calculate the discriminant, which is the part under the square root in the quadratic formula:
$\Delta = b^2 - 4ac$

Plug in the values for $a$ , $b$ , and $c$ to calculate the discriminant:
$\Delta = (-2)^2 - 4 \left(-\frac{1}{2}\right) \left(\frac{7}{10}\right)$

Simplify the discriminant calculation:
$\Delta = 4 - 4 \left(-\frac{1}{2}\right) \left(\frac{7}{10}\right)$

Multiply the terms inside the parentheses:
$\Delta = 4 - (-2) \left(\frac{7}{10}\right)$

Multiply the numbers:
$\Delta = 4 - (-\frac{14}{10})$

Simplify the expression by converting the subtraction of a negative to addition:
$\Delta = 4 + \frac{14}{10}$

Convert the whole number to a fraction with a denominator of 10 to add the fractions:
$\Delta = \frac{40}{10} + \frac{14}{10}$

Add the fractions:
$\Delta = \frac{40}{10} + \frac{14}{10} = \frac{54}{10}$

Simplify the fraction:
$\Delta = \frac{27}{5}$

Now that we have the discriminant, we can use the quadratic formula to find the roots. We will calculate the positive and negative square roots separately.
First, calculate the positive root using the quadratic formula:
$x_1 = \frac{-b + \sqrt{\Delta}}{2a}$

Plug in the values for $a$ , $b$ , and $\Delta$ to calculate the positive root:
$x_1 = \frac{-(-2) + \sqrt{\frac{27}{5}}}{2 \left(-\frac{1}{2}\right)}$

Simplify the numerator:
$x_1 = \frac{2 + \sqrt{\frac{27}{5}}}{-1}$

Calculate the square root of the fraction:
$\sqrt{\frac{27}{5}} = \frac{\sqrt{27}}{\sqrt{5}} = \frac{3\sqrt{3}}{\sqrt{5}}$

Plug the square root back into the formula for $x_1$ :
$x_1 = \frac{2 + \frac{3\sqrt{3}}{\sqrt{5}}}{-1}$

Simplify the expression by distributing the denominator:
$x_1 = -2 - \frac{3\sqrt{3}}{\sqrt{5}}$

Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{5}$ :
$x_1 = -2 - \frac{3\sqrt{3}\sqrt{5}}{5}$

Simplify the expression:
$x_1 = -2 - \frac{3\sqrt{15}}{5}$

Now, calculate the negative root using the quadratic formula:
$x_2 = \frac{-b - \sqrt{\Delta}}{2a}$

Plug in the values for $a$ , $b$ , and $\Delta$ to calculate the negative root:
$x_2 = \frac{-(-2) - \sqrt{\frac{27}{5}}}{2 \left(-\frac{1}{2}\right)}$

Simplify the numerator:
$x_2 = \frac{2 - \frac{3\sqrt{3}}{\sqrt{5}}}{-1}$

Simplify the expression by distributing the denominator:
$x_2 = -2 + \frac{3\sqrt{3}}{\sqrt{5}}$

Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{5}$ :
$x_2 = -2 + \frac{3\sqrt{3}\sqrt{5}}{5}$

Simplify the expression:
$x_2 = -2 + \frac{3\sqrt{15}}{5}$