Systems of equations of the second degree

Algebra, grade 9 Lesson 24 of 26Learning

Systems of equations of the second degree

A system of equations is two equations with two unknowns, $\displaystyle x$ and $\displaystyle y$ , that must hold together. A solution of the system is a pair $\displaystyle (x,y)$ that satisfies both equations. When one of the equations has $\displaystyle x^{2}$ in it, it is a system of the second degree:

\begin{cases}y=x^{2}-1\\ y=x+1\end{cases}

The graphical meaning: intersection points

Each equation is a graph: $\displaystyle {y}={x^{2}-1}$ is a parabola, $\displaystyle {y}={x+1}$ (dashed) is a line. A pair that satisfies both equations is a point on both graphs – an intersection point. Here they meet at $\displaystyle (-1,0)$ and $\displaystyle (2,3)$ .

\displaystyle y=x^{2}-1

\displaystyle y=x+1

\displaystyle A=(-1,0)

\displaystyle B=(2,3)

Solving by substitution

A solution is a pair. Find

\displaystyle y

for each

\displaystyle x

, and write both pairs.

At an intersection point both graphs have the same $\displaystyle y$ , so the two expressions for $\displaystyle y$ are equal:

\begin{aligned}x^{2}-1&=x+1\\ x^{2}-x-2&=0\\ (x-2)(x+1)&=0\end{aligned}

$\displaystyle {x}={2}$ or $\displaystyle {x}={-1}$ . The $\displaystyle y$ of each comes from the simpler equation, $\displaystyle {y}={x+1}$ : for $\displaystyle {x}={2}$ , $\displaystyle {y}={3}$ ; for $\displaystyle {x}={-1}$ , $\displaystyle {y}={0}$ . The solutions: $\displaystyle (2,3)$ and $\displaystyle (-1,0)$ . Check in the other equation: $\displaystyle {2^{2}-1}={3}$ , $\displaystyle {(-1)^{2}-1}={0}$ .

Isolating

\displaystyle y

in the line’s equation

First isolate $\displaystyle y$ in the line’s equation. In the system $\displaystyle {x+y}={5}$ , $\displaystyle {y}={x^{2}-1}$ : $\displaystyle {y}={5-x}$ , so $\displaystyle {x^{2}-1}={5-x}$ , $\displaystyle {x^{2}+x-6}={0}$ , $\displaystyle {(x+3)(x-2)}={0}$ . For $\displaystyle {x}={2}$ , $\displaystyle {y}={3}$ ; for $\displaystyle {x}={-3}$ , $\displaystyle {y}={8}$ . The solutions: $\displaystyle (2,3)$ and $\displaystyle (-3,8)$ .

Exercise 1

Here are the graphs of $\displaystyle {y}={x^{2}-2x-3}$ and $\displaystyle {y}={x-3}$ (dashed).
Mark the intersection points on the graph.
Solve the system algebraically, and compare with the points you marked.

\displaystyle y=x^{2}-2x-3

\displaystyle y=x-3

The number of intersection points

A parabola and a line meet at two points, at one point or not at all – as many as the solutions of the quadratic equation we get.

$\displaystyle {y}={x^{2}}$ and $\displaystyle {y}={2x-1}$ (dashed): $\displaystyle {x^{2}-2x+1}={0}$ , $\displaystyle {(x-1)^{2}}={0}$ – one solution, $\displaystyle {x}={1}$ . The line touches the parabola at $\displaystyle (1,1)$ .
$\displaystyle {y}={x^{2}}$ and $\displaystyle {y}={x-2}$ : $\displaystyle {x^{2}-x+2}={0}$ – under the root $\displaystyle {1-8}={-7}$ , no solution. The line passes below the parabola.

\displaystyle y=x^{2}

\displaystyle y=2x-1

\displaystyle (1,1)

\displaystyle y=x-2

A system of two parabolas

Two parabolas are solved the same way – equate the two expressions for $\displaystyle y$ . For $\displaystyle {y}={x^{2}-4}$ and $\displaystyle {y}={-x^{2}+2x}$ (dashed): $\displaystyle {x^{2}-4}={-x^{2}+2x}$ , $\displaystyle {2x^{2}-2x-4}={0}$ , $\displaystyle {x^{2}-x-2}={0}$ , so $\displaystyle {x}={2}$ or $\displaystyle {x}={-1}$ . The intersection points: $\displaystyle (2,0)$ and $\displaystyle (-1,-3)$ .

\displaystyle y=x^{2}-4

\displaystyle y=-x^{2}+2x

\displaystyle (2,0)

\displaystyle (-1,-3)

Exercise 2

Solve the systems.

Exercise 3

How many intersection points does the parabola $\displaystyle {y}={x^{2}}$ have with each of the lines $\displaystyle {y}={4}$ , $\displaystyle {y}={0}$ and $\displaystyle {y}={-1}$ ?
Draw the parabola and the three lines.
Solve the three systems.

Formulas: algebra, grade 9

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The laws of powers

Multiplying powers with the same base
$\displaystyle a^{m}\cdot a^{n}$ $\displaystyle =$ $\displaystyle a^{m+n}$
Dividing powers with the same base
$\displaystyle a^{m}:a^{n}$ $\displaystyle =$ $\displaystyle a^{m-n}$ $\displaystyle a\neq0$
A power of a power
$\displaystyle \left(a^{m}\right)^{n}$ $\displaystyle =$ $\displaystyle a^{m\cdot n}$
A power of a product
$\displaystyle (a\cdot b)^{n}$ $\displaystyle =$ $\displaystyle a^{n}\cdot b^{n}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{n}$ $\displaystyle =$ $\displaystyle \frac{a^{n}}{b^{n}}$ $\displaystyle b\neq0$
The power zero
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$

The laws of square roots

The root of a product
$\displaystyle \sqrt{a\cdot b}$ $\displaystyle =$ $\displaystyle \sqrt{a}\cdot\sqrt{b}$ $\displaystyle a\ge0,\ b\ge0$
The root of a quotient
$\displaystyle \sqrt{\frac{a}{b}}$ $\displaystyle =$ $\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle a\ge0,\ b>0$
The square of a root
$\displaystyle \left(\sqrt{a}\right)^{2}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$
The root of a square
$\displaystyle \sqrt{a^{2}}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$

The extended distributive law and the short multiplication formulas

The extended distributive law
$\displaystyle (a+b)(c+d)$ $\displaystyle =$ $\displaystyle ac+ad+bc+bd$
The square of a sum
$\displaystyle (a+b)^{2}$ $\displaystyle =$ $\displaystyle a^{2}+2ab+b^{2}$
The square of a difference
$\displaystyle (a-b)^{2}$ $\displaystyle =$ $\displaystyle a^{2}-2ab+b^{2}$
The difference of squares
$\displaystyle (a+b)(a-b)$ $\displaystyle =$ $\displaystyle a^{2}-b^{2}$

The parabola

Standard form
$\displaystyle y=ax^{2}+bx+c$
Vertex form – the vertex is $\displaystyle (b,k)$
$\displaystyle y=a(x-b)^{2}+k$
Product form – it meets the $\displaystyle x$ -axis at $\displaystyle x_1$ and $\displaystyle x_2$
$\displaystyle y=a(x-x_1)(x-x_2)$
The vertex’s $\displaystyle x$ , in standard form
$\displaystyle x=-\frac{b}{2a}$

The quadratic equation

The roots of $\displaystyle {ax^{2}+bx+c}={0}$ , when $\displaystyle a\neq0$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$