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Quadratic Inequalities 2Quadratic Inequalities 2
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Question 1 of 4
1. Question
Solve for `x``3x^2-3<0`Hint
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The difference of two squares, `a^2-b^2`, can be factored as the sum and and difference of `a` and `b` `(a+b)(a-b)`First, change the inequality sign into an equal sign and find the `x` values`3x^2-3` `=` `0` `3(x^2-1)` `=` `0` Factor out `3` `3(x+1)(x-1)` `=` `0` Difference of two squares `x+1` `=` `0` `x+1` `-1` `=` `0` `-1` `x` `=` `-1` `x-1` `=` `0` `x-1` `+1` `=` `0` `+1` `x` `=` `1` Mark these `2` points on the `x` axis.
Next, substitute `x=0` to the function to get the `y` intercept`y` `=` `3x^2-3` `y` `=` `3(0)^2-3` Substitute `x=0` `y` `=` `0-3` `y` `=` `-3` Mark this point on the `y` axis.
Form a parabola by connecting the pointsSince we are looking for `y``<``0`, the values are below the `x` axis
Hence, `-1``<``x``<``1``-1``<``x``<``1` -
Question 2 of 4
2. Question
Solve for `x`:`2x^2+3x-7≥0`Hint
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The Quadratic Formula
$$x=\frac {-\color{#9a00c7}{b} \pm \sqrt {\color{#9a00c7}{b}^2-4\color{#00880A}{a}\color{#007DDC}{c}} }{2 \color{#00880A}{a}}$$First, replace the inequality with an equal sign and solve for `x` using the Quadratic Formula`2x^2+3x-7=0``a=2` `b=3` `c=-7``x` `=` $$\frac {-\color{#9a00c7}{b} \pm \sqrt {\color{#9a00c7}{b}^2-4\color{#00880A}{a}\color{#007DDC}{c}} }{2 \color{#00880A}{a}}$$ Quadratic Formula `=` $$\frac {- \color{#9a00c7}{3} \pm \sqrt {\color{#9a00c7}{3}^2-4\color{#00880A}{(2)}\color{#007DDC}{(-7)}} }{2 \color{#00880A}{(2)}}$$ Plug in the values of `a, b` and `c` `=` $$\frac {-3 \pm \sqrt {9 +56} }{4}$$ `=` $$\frac {-3 \pm \sqrt {65} }{4}$$ Write each root individually$$x_1$$ `=` $$\frac {-3 + \sqrt {65} }{4}$$ `=` $$1.266$$ $$x_2$$ `=` $$\frac {-3 – \sqrt {65} }{4}$$ `=` $$-2.766$$ Mark these two points on the `x` axis
Next, find the `y` intercept by substituting `x=0``y` `=` `2x^2+3x-7` `y` `=` `2(0)^2+3(0)-7` Substitute `x=0` `y` `=` `0-0-7` `y` `=` `-7` Mark this on the `y` axis
Form a parabola by connecting the pointsSince we are looking for `y≥0`, the values are on or above the `x` axis
Hence, `x≤-2.766` and `x≥1.266``x≤-2.766` and `x≥1.266` -
Question 3 of 4
3. Question
Graph the inequality:`y``>``x^2-3x-4`Hint
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Remember the following notations when graphing inequalities.Symbol Solid / Dotted `<` Dotted Line `>` Dotted Line `≤` Solid Line `≥` Solid Line First, equate the function to `0` and solve for `x` by factoring`y` `>` `x^2-3x-4` `0` `=` `x^2-3x-4` `x^2-3x-4` `=` `0` 
`(x-4)(x+1)` `=` `0` `x-4` `=` `0` `x-4` `+4` `=` `0` `+4` `x` `=` `4` `x+1` `=` `0` `x+1` `-1` `=` `0` `-1` `x` `=` `-1` Mark these `2` points on the `x` axis
Next, find the `y` intercept by substituting `x=0``y` `=` `x^2-3x-4` `y` `=` `(0)^2-3(0)-4` Substitute `x=0` `y` `=` `0-0-4` `y` `=` `-4` Mark this on the `y` axis
Now, connect the points to form a parabolaRemember to use a dotted line because of the `>` sign
To determine which region to shade, test the origin by substituting `(0,0)` to the original function`x=0``y=0``y` `>` `x^2-3x-4` `0` `>` `(0)^2-3(0)-4` Substitute values `0` `>` `0-0-4` `0` `>` `-4` This is true, which means the region that includes the origin must be shaded
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Question 4 of 4
4. Question
Graph the system of inequalities:`y≥x^2-4x+3``y``<``4-x^2`Hint
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Remember the following notations when graphing inequalities.Symbol Solid / Dotted `<` Dotted Line `>` Dotted Line `≤` Solid Line `≥` Solid Line Axis of Symmetry
$$x=\frac{-\color{#9a00c7}{b}}{2\color{#00880A}{a}}$$First, graph the first inequalityStart by equating the function to `0` and solving for `x` by factoring`x^2-4x+3` `=` `0` 
`(x-3)(x+1)` `=` `0` `x-3` `=` `0` `x-3` `+3` `=` `0` `+3` `x` `=` `3` `x+1` `=` `0` `x+1` `-1` `=` `0` `-1` `x` `=` `-1` Mark these `2` points on the `x` axis
Next, find the axis of symmetry`y=x^2-4x+3``a=1` `b=-4` `c=3``x` `=` $$\frac{-\color{#9a00c7}{b}}{2\color{#00880A}{a}}$$ Axis of Symmetry `x` `=` $$\frac{-\color{#9a00c7}{(-4)}}{2\color{#00880A}{(1)}}$$ Substitute values `x` `=` `4/2` `x` `=` `2` 
Substitute `x=2` to the equation to find the value of `y` for the vertex`y` `=` `x^2-4x+3` `y` `=` `2^2-4(2)+3` Substitute `x=2` `y` `=` `4-8+3` `y` `=` `-1` This means that the vertex is at `(2,-1)`
Find the `y` intercept by substituting `x=0``y` `=` `x^2-4x+3` `y` `=` `0^2-4(0)+3` Substitute `x=0` `y` `=` `0-0+3` `y` `=` `3` 
Now, connect the points to form a parabolaRemember to use a solid line because of the `≥` sign
To determine which region to shade, test a point by substituting `(2,0)` to the original function`x=2``y=0``y` `≥` `x^2-4x+3` `0` `≥` `2^2-4(2)+3` Substitute values `0` `≥` `4-8+3` `0` `≥` `-1` This is true, which means the region that covers `(2,0)` must be shaded
This time, graph the second inequalityStart by equating the function to `0` and solving for `x` by factoring`4-x^2` `=` `0` `(2-x)(2+x)` `=` `0` `2-x` `=` `0` `2-x` `+x` `=` `0` `+x` `2` `=` `x` `x` `=` `2` `2+x` `=` `0` `2+x` `-x` `=` `0` `-x` `2` `=` `-x` `x` `=` `-2` Mark these `2` points on the `x` axis
Next, find the axis of symmetry`y=4-x^2``y=-x^2+4``a=-1` `b=0` `c=4``x` `=` $$\frac{-\color{#9a00c7}{b}}{2\color{#00880A}{a}}$$ Axis of Symmetry `x` `=` $$\frac{-\color{#9a00c7}{0}}{2\color{#00880A}{(-1)}}$$ Substitute values `x` `=` `0` The axis of symmetry is at `x=0` or the `y` axisSubstitute `x=0` to the equation to find the value of `y` for the vertex`y` `=` `4-x^2` `y` `=` `4-0^2` Substitute `x=0` `y` `=` `4` This means that the vertex is at `(0,4)`Since this point lies on the `y` axis, it is also the `y` intercept
Now, connect the points to form a parabolaRemember to use a dotted line because of the `<` sign
To determine which region to shade, test the origin by substituting `(0,0)` to the original function`x=0``y=0``y` `<` `4-x^2` `0` `<` `4-0^2` Substitute values `0` `<` `4` This is true, which means the region that covers `(0,0)` must be shaded
Finally, highlight the overlapping region of the two inequalities
Quizzes
- Solve Quadratics by Factoring
- The Quadratic Formula
- Completing the Square 1
- Completing the Square 2
- Intro to Quadratic Functions (Parabolas) 1
- Intro to Quadratic Functions (Parabolas) 2
- Intro to Quadratic Functions (Parabolas) 3
- Graph Quadratic Functions in Standard Form 1
- Graph Quadratic Functions in Standard Form 2
- Graph Quadratic Functions by Completing the Square
- Graph Quadratic Functions in Vertex Form
- Write a Quadratic Equation from the Graph
- Write a Quadratic Equation Given the Vertex and Another Point
- Quadratic Inequalities 1
- Quadratic Inequalities 2
- Quadratics Word Problems 1
- Quadratics Word Problems 2
- Quadratic Identities
- Graphing Quadratics Using the Discriminant
- Positive and Negative Definite
- Applications of the Discriminant 1
- Applications of the Discriminant 2
- Combining Methods for Solving Quadratic Equations