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Quadratic Inequalities 2Quadratic Inequalities 2
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Question 1 of 4
1. Question
Solve for `x``3x^23<0`Hint
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The difference of two squares, `a^2b^2`, can be factored as the sum and and difference of `a` and `b` `(a+b)(ab)`First, change the inequality sign into an equal sign and find the `x` values`3x^23` `=` `0` `3(x^21)` `=` `0` Factor out `3` `3(x+1)(x1)` `=` `0` Difference of two squares `x+1` `=` `0` `x+1` `1` `=` `0` `1` `x` `=` `1` `x1` `=` `0` `x1` `+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^23` `y` `=` `3(0)^23` Substitute `x=0` `y` `=` `03` `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` axisHence, `1``<``x``<``1``1``<``x``<``1` 
Question 2 of 4
2. Question
Solve for `x`:`2x^2+3x7≥0`Hint
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The Quadratic Formula
$$x=\frac {\color{#9a00c7}{b} \pm \sqrt {\color{#9a00c7}{b}^24\color{#00880A}{a}\color{#007DDC}{c}} }{2 \color{#00880A}{a}}$$First, replace the inequality with an equal sign and solve for `x` using the Quadratic Formula`2x^2+3x7=0``a=2` `b=3` `c=7``x` `=` $$\frac {\color{#9a00c7}{b} \pm \sqrt {\color{#9a00c7}{b}^24\color{#00880A}{a}\color{#007DDC}{c}} }{2 \color{#00880A}{a}}$$ Quadratic Formula `=` $$\frac { \color{#9a00c7}{3} \pm \sqrt {\color{#9a00c7}{3}^24\color{#00880A}{(2)}\color{#007DDC}{(7)}} }{2 \color{#00880A}{(2)}}$$ 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` axisNext, find the `y` intercept by substituting `x=0``y` `=` `2x^2+3x7` `y` `=` `2(0)^2+3(0)7` Substitute `x=0` `y` `=` `007` `y` `=` `7` Mark this on the `y` axisForm a parabola by connecting the pointsSince we are looking for `y≥0`, the values are on or above the `x` axisHence, `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^23x4`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^23x4` `0` `=` `x^23x4` `x^23x4` `=` `0` `(x4)(x+1)` `=` `0` `x4` `=` `0` `x4` `+4` `=` `0` `+4` `x` `=` `4` `x+1` `=` `0` `x+1` `1` `=` `0` `1` `x` `=` `1` Mark these `2` points on the `x` axisNext, find the `y` intercept by substituting `x=0``y` `=` `x^23x4` `y` `=` `(0)^23(0)4` Substitute `x=0` `y` `=` `004` `y` `=` `4` Mark this on the `y` axisNow, connect the points to form a parabolaRemember to use a dotted line because of the `>` signTo determine which region to shade, test the origin by substituting `(0,0)` to the original function`x=0``y=0``y` `>` `x^23x4` `0` `>` `(0)^23(0)4` Substitute values `0` `>` `004` `0` `>` `4` This is true, which means the region that includes the origin must be shaded 
Question 4 of 4
4. Question
Graph the system of inequalities:`y≥x^24x+3``y``<``4x^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^24x+3` `=` `0` `(x3)(x+1)` `=` `0` `x3` `=` `0` `x3` `+3` `=` `0` `+3` `x` `=` `3` `x+1` `=` `0` `x+1` `1` `=` `0` `1` `x` `=` `1` Mark these `2` points on the `x` axisNext, find the axis of symmetry`y=x^24x+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^24x+3` `y` `=` `2^24(2)+3` Substitute `x=2` `y` `=` `48+3` `y` `=` `1` This means that the vertex is at `(2,1)`Find the `y` intercept by substituting `x=0``y` `=` `x^24x+3` `y` `=` `0^24(0)+3` Substitute `x=0` `y` `=` `00+3` `y` `=` `3` Now, connect the points to form a parabolaRemember to use a solid line because of the `≥` signTo determine which region to shade, test a point by substituting `(2,0)` to the original function`x=2``y=0``y` `≥` `x^24x+3` `0` `≥` `2^24(2)+3` Substitute values `0` `≥` `48+3` `0` `≥` `1` This is true, which means the region that covers `(2,0)` must be shadedThis time, graph the second inequalityStart by equating the function to `0` and solving for `x` by factoring`4x^2` `=` `0` `(2x)(2+x)` `=` `0` `2x` `=` `0` `2x` `+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` axisNext, find the axis of symmetry`y=4x^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` `=` `4x^2` `y` `=` `40^2` 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` interceptNow, connect the points to form a parabolaRemember to use a dotted line because of the `<` signTo determine which region to shade, test the origin by substituting `(0,0)` to the original function`x=0``y=0``y` `<` `4x^2` `0` `<` `40^2` Substitute values `0` `<` `4` This is true, which means the region that covers `(0,0)` must be shadedFinally, 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