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Try 9
Explain why the graph of
y=x^(2)-2x+2-p, where
p > 1, intersects the
x-axis at two points.

Try $9$ $\newline$ Explain why the graph of $y=x^{2}-2 x+2-p$ , where $p>1$ , intersects the $x$ -axis at two points.

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Characteristics of quadratic functions: equations

Full solution

Q. Try

9

\newline

Explain why the graph of

y=x^{2}-2 x+2-p

, where

p>1

, intersects the

x

-axis at two points.

Set $y = 0$ : To find where the graph intersects the x-axis, we set $y = 0$ and solve for $x$ . $0 = x^2 - 2x + 2 - p$
Rearrange quadratic equation: Rearrange the equation to find the roots of the quadratic equation. $\newline$ $x^2 - 2x + (2 - p) = 0$
Calculate discriminant: Use the discriminant, $D = b^2 - 4ac$ , to determine the nature of the roots. $\newline$ Here, $a = 1$ , $b = -2$ , and $c = 2 - p$ . $\newline$ $D = (-2)^2 - 4(1)(2 - p)$
Find discriminant value: Calculate the discriminant. $\newline$ $D = 4 - 4(2 - p)$ $\newline$ $D = 4 - 8 + 4p$ $\newline$ $D = 4p - 4$
Substitute $p = 2$ : Since $p > 1$ , let's substitute $p = 2$ to check if the discriminant is positive. $\newline$ $D = 4(2) - 4$ $\newline$ $D = 8 - 4$ $\newline$ $D = 4$
Interpret discriminant result: The discriminant is positive $D = 4$ , which means there are two real roots, so the graph intersects the x-axis at two points.

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