// Numbas version: exam_results_page_options {"name": "Katherine's copy of Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["a1", "a2", "a3", "a4", "b1", "b2", "b3", "b4", "x1", "p1", "p2", "x2", "a", "m"], "parts": [{"marks": 0, "showFeedbackIcon": true, "steps": [{"marks": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "prompt": "

An equation of the form

\n

\$ax^2+bx+c=0\\text{,}\$

\n

\n

can be solved using the quadratic formula

\n

\$x={\\frac {-b\\pm\\sqrt{b^2-4\\times a\\times c}}{2a}}\\text{.}\$

\n

", "type": "information", "variableReplacements": [], "scripts": {}, "showCorrectAnswer": true}], "stepsPenalty": 0, "variableReplacementStrategy": "originalfirst", "prompt": "

$\\simplify{x^2+{a+m}x+{a*m}=0}$

\n

$x_1=$ [[0]]

\n

$x_2=$ [[1]]

\n

$\\simplify{{a1}x^2+{a2}x+{a3}={a4}}$

\n

$x_1=$ [[0]]

\n

$x_2=$ [[1]]

\n

$\\simplify{{b1}x^2+{b2}x+{b3}={b4}x}$

\n

$x_1=$ [[0]]

\n

$x_2=$ [[1]]

", "type": "gapfill", "variableReplacements": [], "scripts": {}, "showCorrectAnswer": true}], "variables": {"p2": {"name": "p2", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "(-(b2-b4)+((b2-b4)^2-4*b1*b3)^(1/2))/(2*b1)"}, "a3": {"name": "a3", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(-30..4 except 0)"}, "p1": {"name": "p1", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "(-(b2-b4)-((b2-b4)^2-4*b1*b3)^(1/2))/(2*b1)"}, "b4": {"name": "b4", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(-5..5)"}, "b2": {"name": "b2", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(20..35 except a2)"}, "b3": {"name": "b3", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(-30..6 except a3)"}, "a2": {"name": "a2", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(9..30 except a1)"}, "a": {"name": "a", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(15..20)"}, "m": {"name": "m", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(9..a-1)"}, "b1": {"name": "b1", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(2..10 except a1)"}, "x2": {"name": "x2", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "(-(a2)+((a2)^2-4*a1*(a3-a4))^(1/2))/(2*a1)"}, "a1": {"name": "a1", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(2..9)"}, "b": {"name": "b", "templateType": "anything", "group": "part 2", "description": "", "definition": "2c^0.5+a1^2"}, "x1": {"name": "x1", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "(-(a2)-((a2)^2-4*a1*(a3-a4))^(1/2))/(2*a1)"}, "c": {"name": "c", "templateType": "anything", "group": "part 2", "description": "", "definition": "random(-10..4 except 0)^2"}, "a4": {"name": "a4", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(1..15)"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "name": "Katherine's copy of Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$", "rulesets": {}, "statement": "

Use the quadratic formula to calculate values for $x$ in these equations. Input the possible values as $x_1$ and $x_2$, where $x_1<x_2$.

", "extensions": [], "variable_groups": [{"name": "part 2", "variables": ["b", "c"]}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Apply the quadratic formula to find the roots of a given equation. The quadratic formula is given in the steps if the student requires it.

\n

\$x={\\frac {-b\\pm\\sqrt{b^2-4\\times a\\times c}}{2a}}\\text{.}\$

\n

#### a)

\n

From the equation, we can read off values for $a$, $b$ and $c$:

\n

\\\begin{align} a&=1\\text{,}\\\\ b&=\\var{a+m}\\text{,}\\\\ c&=\\var{a*m} \\text{.} \\end{align}\

\n

Substituting these values into the quadratic formula,

\n

\$x = \\frac {-\\var{a+m}\\pm\\sqrt{\\var{a+m}^2-4\\times \\var{a*m}}}{2}\\text{.}\$

\n

Note the $\\pm$ symbol in the formula. This means there are two solutions: one using $+$, the other using $-$.

\n

The two solutions are

\n

\\\begin{align} x_1&=\\var{m}\\text{,}\\\\ x_2&=\\var{a}\\text{.} \\end{align}\

\n

#### b)

\n

Note that the right-hand side of the given equation is not zero. We need to rewrite it in the form $ax^2+bx+c=0$:

\n

\\\begin{align} \\simplify{{a1}x^2+{a2}x+{a3}}&=\\var{a4}\\\\ \\simplify{{a1}x^2+{a2}x+{a3-a4}}&=0\\text{.} \\end{align}\

\n

Then we can read off values for $a$, $b$ and $c$:

\n

\\\begin{align} a&=\\var{a1}\\\\ b&=\\var{a2}\\\\ c&=\\var{a3-a4} \\text{.} \\end{align}\

\n

We can now substitute these values into the quadratic formula:

\n

\$x = {\\frac {-\\var{a2}\\pm\\sqrt{\\var{a2}^2-4\\times \\var{a1}\\times \\var{a3-a4}}}{2\\times\\var{a1}}}\\text{.}\$

\n

So the two solutions are

\n

\\\begin{align} x_1&=\\var{dpformat(x1,2)}\\\\ x_2&=\\var{dpformat(x2,2)}\\text{.} \\end{align}\

\n

#### c)

\n

We first rearrange our equation into the form $ax^2+bx+c=0$:

\n

\\\begin{align} \\simplify{{b1}x^2+{b2}x+{b3}}&=0=\\var{b4}x\\\\ \\simplify{{b1}x^2+{b2-b4}x+{b3}}&=0\\text{.} \\end{align}\

\n

We can then read off the values for $a, b$ and $c$, which are

\n

\\\begin{align} a&=\\var{b1}\\text{,}\\\\ b&=\\var{b2-b4}\\text{,}\\\\ c&=\\var{b3}\\text{.} \\end{align}\

\n

Substituting these values into the quadratic formula,

\n

\$x = {\\frac {-\\var{b2-b4}\\pm\\sqrt{\\var{b2-b4}^2-4\\times \\var{b1}\\times \\var{b3}}}{2\\times\\var{b1}}},\$

\n

we obtain solutions

\n

\\\begin{align} x_1&=\\var{dpformat(p1,2)}\\text{,}\\\\ x_2&=\\var{dpformat(p2,2)}\\text{.} \\end{align}\

", "tags": [], "preamble": {"css": "", "js": ""}, "functions": {}, "type": "question", "contributors": [{"name": "Katherine Tomlinson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1652/"}]}]}], "contributors": [{"name": "Katherine Tomlinson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1652/"}]}