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Write a quadratic in standard form with given roots

3/4,-5

pls give me step explanation

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Answer:

To write a quadratic equation in standard form with the given roots of \( \frac{3}{4} \) and -5, you can follow these steps:

1. Start with the factored form of a quadratic equation, which is \(a(x - r_1)(x - r_2) = 0\), where \(a\) is the leading coefficient and \(r_1\) and \(r_2\) are the roots.

2. Plug in the given roots:

\(a\left(x - \frac{3}{4}\right)(x + 5) = 0\)

3. To find the value of \(a\), you can use any point on the quadratic curve. You have two roots, so you can choose either one. Let's use the root \(x = \frac{3}{4}\). Plug in this value and solve for \(a\):

\(a\left(\frac{3}{4} - \frac{3}{4}\right)\left(\frac{3}{4} + 5\right) = 0\)

\(a(0)\left(\frac{27}{4}\right) = 0\)

Since the left side is equal to 0, we can conclude that \(a\) can be any real number.

4. Now, you have your quadratic equation in factored form:

\(a\left(x - \frac{3}{4}\right)(x + 5) = 0\)

5. If you want the quadratic equation in standard form (ax^2 + bx + c = 0), you can expand the equation:

\(a(x - \frac{3}{4})(x + 5) = 0\)

\(a(x^2 + 5x - \frac{3}{4}x - \frac{15}{4}) = 0\)

\(a(x^2 + \frac{17}{4}x - \frac{15}{4}) = 0\)

Now, your quadratic equation in standard form with the given roots \( \frac{3}{4} \) and -5 is:

\(a(x^2 + \frac{17}{4}x - \frac{15}{4}) = 0\)

You can multiply all terms by a constant if you want a specific value for \(a\), but in general, this is the standard form of the quadratic equation with the given roots.

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