Equations and inequalities of the first and second degree

Algebra and calculus, grade 10 (4 units) Lesson 2 of 20Learning

Equations and inequalities of the first and second degree

A first-degree equation

In a first-degree equation the unknown appears only to the first power. Expand the brackets, move the terms with $\displaystyle x$ to one side and the numbers to the other, and divide by the coefficient of $\displaystyle x$ .

3(x-2)=x+4

We expand the brackets: $\displaystyle {3x-6}={x+4}$ .
We subtract $\displaystyle x$ and add 6 on both sides: $\displaystyle {2x}={10}$ .
We divide by 2: $\displaystyle {x}={5}$ .

A quadratic equation

A quadratic equation $\displaystyle {ax^2+bx+c}={0}$ is solved by factoring or by the quadratic formula:

\displaystyle x_{1,2}

\displaystyle =

\displaystyle \frac{-b\pm\sqrt{b^2-4ac}}{2a}

The expression under the root, $\displaystyle b^2-4ac$ , is called the discriminant. It decides how many solutions the equation has:

$\displaystyle b^2-4ac>0$ – two solutions;
$\displaystyle {b^2-4ac}={0}$ – one solution;
$\displaystyle b^2-4ac<0$ – no solution, because a negative number has no square root.

The solutions of $\displaystyle {ax^2+bx+c}={0}$ are the $\displaystyle x$ -coordinates of the points where the parabola $\displaystyle {y}={ax^2+bx+c}$ meets the $\displaystyle x$ -axis. On the board – three parabolas that differ only in the constant term:

\displaystyle y=x^{2}-4x+3

\displaystyle y=x^{2}-4x+4

\displaystyle y=x^{2}-4x+5

$\displaystyle {y}={x^2-4x+3}$ crosses the $\displaystyle x$ -axis twice: $\displaystyle {x^2-4x+3}={0}$ has two solutions, 1 and 3.
$\displaystyle {y}={x^2-4x+4}$ touches the $\displaystyle x$ -axis at one point: $\displaystyle {x^2-4x+4}={0}$ has one solution, 2.
$\displaystyle {y}={x^2-4x+5}$ is wholly above the $\displaystyle x$ -axis: $\displaystyle {x^2-4x+5}={0}$ has no solution.

A system of equations

A system of two equations in two unknowns is solved by substitution: isolate an unknown in one equation and put it into the other. Every solution of the system is a meeting point of the two graphs.

The example below solves the system $\displaystyle {y}={x+2}$ , $\displaystyle {y}={x^2}$ :

We put $\displaystyle {y}={x+2}$ into the second equation: $\displaystyle {x^2}={x+2}$ , that is $\displaystyle {x^2-x-2}={0}$ .
We factor: $\displaystyle {(x-2)(x+1)}={0}$ , so $\displaystyle {x}={2}$ or $\displaystyle {x}={-1}$ .
We put them into $\displaystyle {y}={x+2}$ : the solutions are $\displaystyle (2,4)$ and $\displaystyle (-1,1)$ – the two meeting points of the line and the parabola.

\displaystyle y=x^{2}

\displaystyle y=x+2

\displaystyle (-1,1)

\displaystyle (2,4)

A second-degree inequality

The solution of an equation is single numbers; the solution of an inequality is a range of numbers. A second-degree inequality is solved with the parabola:

solve the equation – these are the points where the parabola crosses the $\displaystyle x$ -axis;
by the sign of $\displaystyle a$ , decide whether the parabola opens upwards or downwards;
read off the graph where the parabola is above the $\displaystyle x$ -axis and where below it.

The example below solves $\displaystyle x^2-2x-3>0$ . The solutions of $\displaystyle {x^2-2x-3}={0}$ are $\displaystyle -1$ and 3, and $\displaystyle {a}={1>0}$ , so the parabola opens upwards:

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

\displaystyle (-1,0)

\displaystyle (3,0)

The parabola is above the $\displaystyle x$ -axis to the left of $\displaystyle -1$ and to the right of 3: the solution is $\displaystyle x<-1$ or $\displaystyle x>3$ .
Between $\displaystyle -1$ and 3 it is below the axis: that is the solution of $\displaystyle x^2-2x-3<0$ , namely $\displaystyle -1<x<3$ .

An inequality between two functions

The solution of $\displaystyle f(x)>g(x)$ is the $\displaystyle x$ -values where the graph of $\displaystyle f$ is above the graph of $\displaystyle g$ . On the board – $\displaystyle {f(x)}={x^2}$ , and the line $\displaystyle {g(x)}={x+2}$ (the dashed one):

\displaystyle f(x)=x^{2}

\displaystyle g(x)=x+2

\displaystyle (-1,1)

\displaystyle (2,4)

The graphs meet where $\displaystyle {x}={-1}$ and where $\displaystyle {x}={2}$ : these are the solutions of $\displaystyle {x^2}={x+2}$ .
To the left of $\displaystyle -1$ and to the right of 2 the parabola is above the line: the solution of $\displaystyle x^2>x+2$ is $\displaystyle x<-1$ or $\displaystyle x>2$ .
Algebra gives the same solution: $\displaystyle x^2-x-2>0$ , and the roots of the parabola $\displaystyle {y}={x^2-x-2}$ are $\displaystyle -1$ and 2.

Exercise 1

Solve the equations.

Exercise 2

Without solving: how many solutions does each equation have?
$\displaystyle {x^2+3x+5}={0}$
$\displaystyle {4x^2-4x+1}={0}$
$\displaystyle {3x^2-x-2}={0}$

\displaystyle {x^2+3x+5}={0}

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle {4x^2-4x+1}={0}

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle {3x^2-x-2}={0}

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

Solve the systems of equations.

Mark on the board the meeting points of $\displaystyle {f(x)}={x^2-3}$ and $\displaystyle {g(x)}={2x}$ .

\displaystyle f(x)=x^{2}-3

\displaystyle g(x)=2x

Solve the system by algebra, and check that you get the same points.

The system in part 4 has no solution.
What does that say about the parabola $\displaystyle {y}={x^2+1}$ and the line $\displaystyle {y}={x-1}$ ?

Exercise 4

Solve the inequalities.

Exercise 5

$\displaystyle {f(x)}={x^2-4x}$
$\displaystyle {g(x)}={-x+4}$

Mark on the board the meeting points of the graphs.

\displaystyle f(x)=x^{2}-4x

\displaystyle g(x)=-x+4

Solve the inequality $\displaystyle f(x)\le g(x)$ : for which values of $\displaystyle x$ is the graph of $\displaystyle f$ not above the graph of $\displaystyle g$ ?

Formulas: algebra and calculus, grade 10 (4 units)

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Algebraic technique

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
$\displaystyle (a+b)(a-b)$ $\displaystyle =$ $\displaystyle a^2-b^2$
Factoring a trinomial – $\displaystyle x_1$ , $\displaystyle x_2$ the roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle ax^2+bx+c$ $\displaystyle =$ $\displaystyle a(x-x_1)(x-x_2)$
The roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
The product of roots
$\displaystyle \sqrt{a}\cdot\sqrt{b}$ $\displaystyle =$ $\displaystyle \sqrt{ab}$ $\displaystyle a\geq0,\ b\geq0$
The quotient of roots
$\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle =$ $\displaystyle \sqrt{\frac{a}{b}}$ $\displaystyle a\geq0,\ b>0$

The properties of a function

An even function – its graph symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – its graph symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The $\displaystyle x$ -coordinate of the vertex of $\displaystyle {y}={ax^2+bx+c}$
$\displaystyle x=\frac{-b}{2a}$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transforming a graph

A shift of $\displaystyle k$ units up ( $\displaystyle k<0$ – down)
$\displaystyle y=f(x)+k$
A shift of $\displaystyle p$ units to the right ( $\displaystyle p<0$ – to the left)
$\displaystyle y=f(x-p)$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$
A stretch ( $\displaystyle |a|>1$ ) or a shrink ( $\displaystyle |a|<1$ )
$\displaystyle y=a\cdot f(x)$

The derivative and the rules of differentiation

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the chord
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
The derivative at a point – the slope of the tangent
$\displaystyle f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}$
The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
The derivative of a constant
$\displaystyle (c)'$ $\displaystyle =$ $\displaystyle 0$
The derivative of a constant multiple
$\displaystyle (c\cdot f(x))'$ $\displaystyle =$ $\displaystyle c\cdot f'(x)$
The derivative of a sum and of a difference
$\displaystyle (f(x)\pm g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\pm g'(x)$
The derivative of a product
$\displaystyle (f(x)\cdot g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a composite function
$\displaystyle (f(g(x)))'$ $\displaystyle =$ $\displaystyle f'(g(x))\cdot g'(x)$

The tangent to a graph

The slope of the tangent at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle m=f'(x_1)$
The equation of the tangent
$\displaystyle y-y_1=m(x-x_1)$

The derivative of a square root function

The derivative of $\displaystyle \sqrt{x}$
$\displaystyle (\sqrt{x})'$ $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x}}$ $\displaystyle x>0$
The derivative of the root of a function
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

Rising, falling and extremum points

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A point that may be an extremum
$\displaystyle f'(x)=0$