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Irina-Kira [14]
3 years ago
15

Solve the quadratic equation 3x^2-5x-7=0

Mathematics
2 answers:
Murljashka [212]3 years ago
8 0

Step-by-step explanation:

3x^{2} -5x -7 = 0 \\ 3x^{2} -5x = 7 \\ x^{2} -\frac{5}{3}x = \frac{7}{3} \\ x^{2} -\frac{5}{3}x +\frac{25}{36} = \frac{7}{3} +\frac{25}{36} \\ (x -\frac{5}{6})^{2} = \frac{84 +25}{36} \\ (x -\frac{5}{6})^{2} = \frac{109}{36} \\ x -\frac{5}{6} = ±\sqrt{\frac{109}{36}} \\ x -\frac{5}{6} = ±\frac{\sqrt{109}}{6} \\ x = ±\frac{\sqrt{109}}{6} +\frac{5}{6}

Equation from the positive root:

x = \frac{\sqrt{109}}{6} +\frac{5}{6} \\ x = \frac{\sqrt{109} +5}{6}

Equation from the negative root:

x = -\frac{\sqrt{109}}{6} +\frac{5}{6} \\ x = \frac{-\sqrt{109} +5}{6}

<h3>Answer:</h3>

x = \frac{\sqrt{109} +5}{6}\\ and x = \frac{-\sqrt{109} +5}{6}\\ satisfy your quadratic equation.

Verdich [7]3 years ago
6 0

Given : A quadratic equation is given to us . The equation is 3x² - 5x - 7 = 0 .

To Find : The roots of the equation .

Solution : Given quadratic equation is 3x²-5x-7=0. So , let's factorise it to get the zeroes of the equation .

⇒ 3x² -5x - 7 = 0.

Here now , use Quadratic formula , of the quadratic equation in standard form of ax² + bx + c = 0.

\boxed{\red{\bf x\:\:=\:\:\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}}}

On substituting respective values ,

⇒ x = -(-5) ± √ (-5)² - 4×3×(-7) / 2 × 3.

⇒x = 5 ± √ [ 25 + 84 ]/ 6 .

⇒ x = 5 ± √ 109 / 6 .

⇒ x = 5 + √109 / 6 , 5 - √109 / 6.

\purple{\underbrace{\underline{\boxed{\red{\tt \purple{\longmapsto} x = \dfrac{5+\sqrt{109}}{6}, \dfrac{5-\sqrt{109}}{6}}}}}}

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A family on vacation records the time in hours) and the distance (in miles) that they travel. The data are shown in the scatter
nata0808 [166]

Answer:

In the first 3.5 hours, the family travelled 175 miles.

Step-by-step explanation:

We have the table:

time (hours)              distance ( miles)

      0                                   0

      1                                   50

       2                                 100

      3                                  150

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      5                                 250

      6                                 300

      7                                 350

      8                                 400

      9                                 450

This seems like a linear relationship, let's write the general line:

y = a*x + b

That represents this situation, y is the distance in miles and x is the time in hours.

In the line equation, a represents the slope and b the y-intercept.

And if the line passes through the points (x₁, y₁) and (x₂, y₂), then the slope of the line is:

a = (y₂ - y₁)/(x₂ - x₁)

In this case we have a lot of points for the line, we can use any pair we want (we actually should use a few different pairs to see that the slope is always the same)

So if we use the first two (0, 0) and (1, 50)

the slope is:

a = (50 - 0)/(1 - 0) = 50

Now, if we use the third and fifth points (4, 200) and (5, 250) now the slope is:

a = (250 - 200)/(5 - 4) = 50/1 = 50

So we can conclude that the slope of the equation is a = 50

Then the equation is something like:

y = 50*x + b

To find the value of b, we can use any of the points of the line, for example (0, 0)

This means that when x = 0, y also must be 0, then:

0 = 50*0 + b

0 = b

Then the equation is:

y = 50*x

Now, to answer how many miles the family travelled in the first 3.5 hours, we just need to replace x by 3.5 in the equation:

y = 50*3.5 = 175

In the first 3.5 hours, the family travelled 175 miles.

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<h3>How to determine the similarity statement</h3>

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DE = \sqrt{(0 -2)^2 + (0 -0)^2} = 2

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Read more about similar triangles at:

brainly.com/question/14285697

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