3.2 Quadratic equations (Part 1)

Solve the following equations. Write your answer in ascending order, and do not use decimal.

(a) [math]x^2-3x-4=0[/math]

(b) [math]x^2-13x+42=10x[/math]

If the solutions to the equation [math]x(x-4)=-1[/math] are [math]a\pm\sqrt{b}[/math] where [math]a[/math] and [math]b[/math] are integers, what is the value of [math]b[/math]?

If [math]p[/math] and [math]q[/math] are the solutions to the equation [math](x-2)(x+3)=1[/math] and [math]p>q[/math] , what is the value of [math]p-q[/math]?

If [math]a\pm\sqrt{b}i[/math] are the solutions to the equation [math]-100x^2-400x=1000[/math], what is the value of [math]b[/math]?

How many real solutions does the equation [math](x-3)(2x+7)=2(2x+7)(2x+1)[/math] have?

How many real solutions does the equation [math](x-2)(x+3)=-(x-2)(x+1)[/math] have?

Solve the equation [math]-2x^2-4x=2[/math]. If you have two solutions, write the bigger one.

If [math]a\pm\sqrt{b}[/math] are the solutions to the equation [math](9-x)^2=2[/math], what is the value of [math]a+b[/math]?

How many common solutions to the equations [math](x^2-4x)^2(x^2-2x-8)^2=0[/math] and [math](x^2-4x)^2+(x^2-2x-8)^2=0[/math] have?

Solve the following equations. If you have two answers. Write the smaller one.

(a) [math]9x(2x-1)=2[/math]

(b) [math]2x^2+7x-15=0[/math]

Solve the following equations. If you have two answers. Write the smaller one.

(a) [math]x^2=3x[/math]

(b) [math]\dfrac{1}{x^2}-\dfrac{8}{x}-9=0[/math]

Solve the following equations. If you have two answers. Write the smaller one.

(a) [math]x(2x+1)-(2x+1)=0[/math]

(b) [math]|x^2-6x+8|+|x^2-x-12|=0[/math]

Solve the equations [math](25x+42)^2-(7x-36)^2=0[/math]. If you have two answers. Write the smaller one. (Do not expand!)

Consider a quadratic equation [math]x^2+bx+c=0[/math] where [math]b[/math] and [math]c[/math] are integers. If [math]3\pm\sqrt{5}[/math] are the solutions to the equation, find the value of [math]b+c[/math].

Which of the following quadratic functions cannot be written in factor form?

Solve [math]6x-37\sqrt{x}+56=0[/math]. If you have two answers, write the bigger one.

Solve [math]w^2+14=0[/math]

Solve [math]3k-10\sqrt{k}+8=0[/math]. If you have two answers, write the bigger one.

How many real solutions does the equation [math](x^2-1)^2+(x^2-1)-12=0[/math] have?

For which of the following equations is the product of its roots NOT a negative number?

(Golden ratio) Consider a rectangle with width 1 and length [math]r[/math]. After we cut a square with length 1 off the original rectangle, we get a small rectangle which is similar to the original rectangle. If [math]r=a+b\sqrt{5}[/math] where [math]a[/math] and [math]b[/math] are rational numbers, what is the value of [math]a+b[/math]?