### 5.3 (OPTIONAL) Finding Quadratics with Specific Zeros

Lesson materials located below the video overview.

Some textbooks want student to do one more thing in writing down quadratic formulas. To set things up, consider the challenge:

Write down a quadratic that crosses the x-axis at 2 and at 5.

Since we are given two obvious interesting $$x$$-values, clearly

$$y=a(x-2)(x-5)$$

does the trick. We can choose any value we like for the steepness $$a$$:

$$y=3(x-2)(x-5)$$  works.
$$y=-5(x-2)(x-5)$$ works.
$$y==\frac{\sqrt{\pi}}{336+\sqrt{2}}(x-2)(x-5)$$ works.

And so on.

Recall that all solutions to the practice problems appear in the COMPANION GUIDE to this QUADRATICS course.

PRACTICE 115:

a) Write down a quadratic with $$x=677$$ and $$x=-6677$$ as zeros.

b) Write down a quadratic that crosses the $$x$$-axis at $$0$$ and at $$-3$$.

c) Write down a quadratic that touches the $$x$$-axis only at $$x=40$$.

Here’s a slightly trickier question:

EXAMPLE 116: Write down a quadratic $$y=ax^{2}+bx+c$$ with $$a$$, $$b$$ and $$c$$ each an integer that crosses the $$x$$-axis at $$x=\frac{4}{3}$$ and at $$x=\frac{50}{7}$$.

Answer: Certainly $$y=\left(x-\frac{4}{3}\right) \left(x-\frac{50}{7}\right)$$ crosses the x-axis at these values but, when expanded, the coefficients involved won’t be integers. To “counteract” the denominators of a 3 and a 7, what if we inserted the number 21 into this formula?

$$y=21\left(x-\frac{4}{3}\right) \left(x-\frac{50}{7}\right)$$.

This is certainly still a quadratic with the desired zeros. And if we expand this slightly we see:

$$y=3\cdot 7 \left(x-\frac{4}{3}\right) \left(x-\frac{50}{7}\right)$$

$$y=3\left(x-\frac{4}{3}\right)\cdot 7 \left(x-\frac{50}{7}\right)$$

$$y=(3x-4)(7x-50)$$

When this is expanded fully, it is clear now that all the coefficients involved will be integers. □

PRACTICE 117: Write examples of quadratics, involving only integer coefficients, with the following zeros:

a) $$x=\frac{1}{2}$$ and $$x=\frac{1}{3}$$.

b) $$x=-\frac{90}{13}$$ and $$x=\frac{19}{2}$$.

c)  $$x=5$$ and $$x=\frac{3}{7}$$.

d) Just one zero at $$x=14\frac{3}{11}$$.

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