"What condition must the discriminant $\\Delta$ satisfy for a quadratic to have two distinct real roots?","back":"$$\\Delta > 0$","points":1},{"id":"11371329-a61b-40f5-849b-d9852d5ba5f2","cardType":"basic","front":"What condition must the discriminant $\\Delta$ satisfy for a quadratic to have one repeated root (equal roots)?","back":"$$\\Delta = 0$","points":1},{"id":"6712a9f4-9300-4f5f-8a7f-86094539deb4","cardType":"basic","front":"What condition must the discriminant $\\Delta$ satisfy for a quadratic to have no real roots?","back":"$$\\Delta < 0$","points":1},{"id":"85601975-ec5b-4573-ae6b-7689806e5d59","cardType":"cloze","front":"If a quadratic graph touches the $x$-axis at exactly one point, its discriminant is $\\text{[...]}$.","back":"If a quadratic graph touches the $x$-axis at exactly one point, its discriminant is $\\boxed{\\Delta = 0}$.","points":null},{"id":"1f9d45bb-61f4-4133-9d43-b6a9f657809b","cardType":"basic","front":"In the equation $5x^2+(k-3)x-2=0$, identify the coefficients $a$, $b$, and $c$.","back":"

$a = 5$
$b = k-3$
$c = -2$

","points":2},{"id":"bed8ad72-53a5-4af5-a620-2bfa000d4ade","cardType":"basic","front":"Identify the common mistake when identifying coefficients from an equation like $x^2 - (k+1)x + 7 = 0$.","back":"Forgetting that the minus sign is part of the coefficient ($b = -(k+1)$).","points":1},{"id":"754dfb76-1bde-4e6d-a377-b8b4e29c9670","cardType":"basic","front":"Why must you check if the coefficient $a$ can equal zero when solving for a parameter?","back":"If $a=0$, the equation becomes linear and the discriminant test is invalid.","points":2},{"id":"2629e59f-b393-4822-b2e7-c1835cdd9364","cardType":"cloze","front":"The discriminant comes from the part of the quadratic formula $\\text{[...]}$; if this value is negative, the roots are not real.","back":"The discriminant comes from the part of the quadratic formula $\\boxed{\\sqrt{b^2-4ac}}$; if this value is negative, the roots are not real.","points":null},{"id":"4e9b7f2a-d136-4b3d-8572-bae114312955","cardType":"basic","front":"How do you find the range of $k$ if $k^2 < 24$?","back":"$$-2\\sqrt{6} < k < 2\\sqrt{6}$","points":2},{"id":"9b88022d-277c-4f1e-b072-a5214fed9b82","cardType":"basic","front":"What is the first step to determine if a line and a curve intersect using the discriminant?","back":"Set the expressions for $y$ equal to each other and rearrange into the form $ax^2+bx+c=0$.","points":2},{"id":"d4b6b233-1d16-497c-a58b-8849a045acc4","cardType":"basic","front":"What discriminant condition describes a line that is a tangent to a curve?","back":"$$\\Delta = 0$ (the line touches the curve at one point).","points":1},{"id":"4e2a60a7-7ba1-4766-a622-bb2c2a715f46","cardType":"cloze","front":"If the discriminant of the equation formed by a curve and a line is negative, the curve and line [ ].","back":"If the discriminant of the equation formed by a curve and a line is negative, the curve and line do not intersect.","points":null},{"id":"7d7131a9-fc59-442f-a54a-e41ac24ae04a","cardType":"basic","front":"A circle and a line have one point of intersection. After substituting the line into the circle equation, what is the value of the discriminant?","back":"$$0$","points":1},{"id":"1a83e511-342d-46df-becc-201c631c7d5d","cardType":"cloze","front":"In the equation $(r+2)x^2+5x+r=0$, the value $\\text{[...]}$ must be excluded from the range of possible solutions for two distinct roots.","back":"In the equation $(r+2)x^2+5x+r=0$, the value $\\boxed{r = -2}$ must be excluded from the range of possible solutions for two distinct roots.","points":null},{"id":"e75a38dd-e570-4ab7-bdcb-eb963ce00048","cardType":"basic","front":"How do you solve a quadratic inequality in a parameter (e.g.,

2.4 The Discriminant Flashcards

2.4 The Discriminant Flashcards

2.4 The Discriminant Flashcards

2.4 The Discriminant Flashcards