Question

unit 2a - quadratic functions

1. solve by graphing.

a) $t^2 - 5t - 150 = 0$ b) $h^2 - 400 = 0$ c) $0 = x^2 + 0.6x - 0.05$ d)$5y^2 + 3y + 100 = 0$

1. for what values of $m$ would the equation $x^2 + 8x + m = 0$ have

a) one real root or two equal real roots? b) two real distinct roots?
c) no real roots?

Explanation:

Problem 5

For each quadratic equation $ax^2+bx+c=0$, the roots correspond to the x-intercepts of the graph $y=ax^2+bx+c$. We can also use the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ to find the roots (which match the graph's intercepts).

a) Step1: Identify a,b,c

$a=1, b=-5, c=-150$

a) Step2: Compute discriminant

$\Delta=(-5)^2-4(1)(-150)=25+600=625$

a) Step3: Calculate roots

$t=\frac{5\pm\sqrt{625}}{2(1)}=\frac{5\pm25}{2}$
$t_1=\frac{5+25}{2}=15$, $t_2=\frac{5-25}{2}=-10$

b) Step1: Identify a,b,c

$a=1, b=0, c=-400$

b) Step2: Compute discriminant

$\Delta=0^2-4(1)(-400)=1600$

b) Step3: Calculate roots

$h=\frac{0\pm\sqrt{1600}}{2(1)}=\frac{\pm40}{2}$
$h_1=20$, $h_2=-20$

c) Step1: Identify a,b,c

$a=1, b=0.6, c=-0.05$

c) Step2: Compute discriminant

$\Delta=(0.6)^2-4(1)(-0.05)=0.36+0.2=0.56$

c) Step3: Calculate roots

$x=\frac{-0.6\pm\sqrt{0.56}}{2(1)}=\frac{-0.6\pm2\sqrt{0.14}}{2}=-0.3\pm\sqrt{0.14}$
$x_1\approx-0.3+0.374=0.074$, $x_2\approx-0.3-0.374=-0.674$

d) Step1: Identify a,b,c

$a=5, b=3, c=100$

d) Step2: Compute discriminant

$\Delta=3^2-4(5)(100)=9-2000=-1991$

d) Step3: Determine root nature

Since $\Delta<0$, no real roots (graph does not cross x-axis)

For quadratic $ax^2+bx+c=0$, use discriminant $\Delta=b^2-4ac$:

• $\Delta=0$: one real (equal) root
• $\Delta>0$: two distinct real roots
• $\Delta<0$: no real roots

Here, $a=1, b=8, c=m$, so $\Delta=8^2-4(1)(m)=64-4m$

a) Step1: Set $\Delta=0$

$64-4m=0$

a) Step2: Solve for m

$4m=64 \implies m=16$

b) Step1: Set $\Delta>0$

$64-4m>0$

b) Step2: Solve inequality

$4m<64 \implies m<16$

c) Step1: Set $\Delta<0$

$64-4m<0$

c) Step2: Solve inequality

$4m>64 \implies m>16$

Answer:

a) $t=15$ and $t=-10$
b) $h=20$ and $h=-20$
c) $x\approx0.074$ and $x\approx-0.674$ (or exact form $x=-0.3\pm\sqrt{0.14}$)
d) No real roots

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Problem 6